On the Unique Games Conjecture (Invited Survey)
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paper_title: On the power of unique 2-prover 1-round games

paper_content:

A 2-prover game is called unique if the answer of one prover uniquely determines the answer of the second prover and vice versa (we implicitly assume games to be one round games). The value of a 2-prover game is the maximum acceptance probability of the verifier over all the prover strategies. We make a conjecture regarding the power of unique 2-prover games, which we call the Unique Games Conjecture.
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paper_title: On the power of unique 2-prover 1-round games

paper_content:

A 2-prover game is called unique if the answer of one prover uniquely determines the answer of the second prover and vice versa (we implicitly assume games to be one round games). The value of a 2-prover game is the maximum acceptance probability of the verifier over all the prover strategies. We make a conjecture regarding the power of unique 2-prover games, which we call the Unique Games Conjecture.
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paper_title: Some optimal inapproximability results

paper_content:

We prove optimal, up to an arbitrary e > 0, inapproximability results for Max-E k-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover.
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paper_title: Limit theorems for polylinear forms

paper_content:

The limit theorems for polylinear forms are obtained. Conditions are found under which the distribution of the polylinear form of many random variables is essentially the same as if all the distributions of arguments were normal.
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paper_title: Inapproximability of NP-complete problems, discrete fourier analysis, and geometry

paper_content:

This article gives a survey of recent results that connect three areas in com- puter science and mathematics: (1) (Hardness of) computing approximate solutions to NP-complete problems. (2) Fourier analysis of boolean functions on boolean hypercube. (3) Certain problems in geometry, especially related to isoperimetry and embeddings between metric spaces.
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paper_title: Optimal algorithms and inapproximability results for every CSP?

paper_content:

Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the following result : If UGC is true, then for every constraint satisfaction problem(CSP) the best approximation ratio is given by a certain simple SDP. Specifically, we show a generic conversion from SDP integrality gaps to UGC hardness results for every CSP. This result holds both for maximization and minimization problems over arbitrary finite domains. Using this connection between integrality gaps and hardness results we obtain a generic polynomial-time algorithm for all CSPs. Assuming the Unique Games Conjecture, this algorithm achieves the optimal approximation ratio for every CSP. Unconditionally, for all 2-CSPs the algorithm achieves an approximation ratio equal to the integrality gap of a natural SDP used in literature. Further the algorithm achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut and Unique Games.
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paper_title: Inapproximability of Combinatorial Optimization Problems

paper_content:

We survey results on the hardness of approximating combinatorial optimization problems.
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paper_title: The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into ℓ1

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In this paper, we disprove a conjecture of Goemans and Linial; namely, that every negative type metric embeds into $\ell_1$ with constant distortion. We show that for an arbitrarily small constant $\delta> 0$, for all large enough $n$, there is an $n$-point negative type metric which requires distortion at least $(\log\log n)^{1/6-\delta}$ to embed into $\ell_1.$ ::: Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot, establishing a previously unsuspected connection between probabilistically checkable proof systems (PCPs) and the theory of metric embeddings. We first prove that the UGC implies a super-constant hardness result for the (non-uniform) Sparsest Cut problem. Though this hardness result relies on the UGC, we demonstrate, nevertheless, that the corresponding PCP reduction can be used to construct an "integrality gap instance" for Sparsest Cut. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of Unique Games. Then we "simulate" the PCP reduction and "translate" the integrality gap instance of Unique Games to an integrality gap instance of Sparsest Cut. This enables us to prove a $(\log \log n)^{1/6-\delta}$ integrality gap for Sparsest Cut, which is known to be equivalent to the metric embedding lower bound.
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paper_title: Noise stability of functions with low influences: Invariance and optimality

paper_content:

In this paper we study functions with low influences on product probability spaces. The analysis of boolean functions with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics. ::: We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known non-linear invariance principles. It has the advantage that its proof is simple and that the error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly ``smoothed''; this extension is essential for our applications to ``noise stability''-type problems. ::: In particular, as applications of the invariance principle we prove two conjectures: the ``Majority Is Stablest'' conjecture from theoretical computer science, which was the original motivation for this work, and the ``It Ain't Over Till It's Over'' conjecture from social choice theory.
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paper_title: L_1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry

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We survey connections between the theory of bi-Lipschitz embeddings and the Sparsest Cut Problem in combinatorial optimization. The story of the Sparsest Cut Problem is a striking example of the deep interplay between analysis, geometry, and probability on the one hand, and computational issues in discrete mathematics on the other. We explain how the key ideas evolved over the past 20 years, emphasizing the interactions with Banach space theory, geometric measure theory, and geometric group theory. As an important illustrative example, we shall examine recently established connections to the the structure of the Heisenberg group, and the incompatibility of its Carnot-Carath\'eodory geometry with the geometry of the Lebesgue space $L_1$.
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paper_title: O(√log n) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems

paper_content:

We give O(√log n)-approximation algorithms for the MIN UNCUT, MIN 2CNF DELETION, DIRECTED BALANCED SEPERATOR, and DIRECTED SPARSEST CUT problems. The previously best known algorithms give an O(log n)-approximation for MIN UNCUT [9], DIRECTED BALANCED SEPERATOR [17], DIRECTED SPARSEST CUT [17], and an O(log n log log n)-approximation for MIN 2CNF DELETION [14].We also show that the integrality gap of an SDP relaxation of the MINIMUM MULTICUT problem is Ω(log n).
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paper_title: Proof verification and the hardness of approximation problems

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We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof” with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [1998] whose verifiers examine a nonconstant number of bits in the proof (though this number is a very slowly growing function of the input length). ::: As a consequence, we prove that no MAX SNP-hard problem has a polynomial time approximation scheme, unless NP = P. The class MAX SNP was defined by Papadimitriou and Yannakakis [1991] and hard problems for this class include vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige et al. [1996] and Arora and Safra [1998] and show that there exists a positive ε such that approximating the maximum clique size in an N-vertex graph to within a factor of Nε is NP-hard.
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paper_title: Probabilistic checking of proofs: a new characterization of NP

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We give a new characterization of NP: the class NP contains exactly those languages <italic>L</italic> for which membership proofs (a proof that an input <italic>x</italic> is in <italic>L</italic>) can be verified probabilistically in polynomial time using <italic>logarithmic</italic> number of random bits and by reading <italic>sublogarithmic</italic> number of bits from the proof. ::: We discuss implications of this characterization; specifically, we show that approximating Clique and Independent Set, even in a very weak sense, is NP-hard.
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paper_title: Interactive proofs and the hardness of approximating cliques

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The contribution of this paper is two-fold. First, a connection is established between approximating the size of the largest clique in a graph and multi-prover interactive proofs. Second, an efficient multi-prover interactive proof for NP languages is constructed, where the verifier uses very few random bits and communication bits. Last, the connection between cliques and efficient multi-prover interaction proofs, is shown to yield hardness results on the complexity of approximating the size of the largest clique in a graph. Of independent interest is our proof of correctness for the multilinearity test of functions.
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paper_title: O(√log n) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems

paper_content:

We give O(√log n)-approximation algorithms for the MIN UNCUT, MIN 2CNF DELETION, DIRECTED BALANCED SEPERATOR, and DIRECTED SPARSEST CUT problems. The previously best known algorithms give an O(log n)-approximation for MIN UNCUT [9], DIRECTED BALANCED SEPERATOR [17], DIRECTED SPARSEST CUT [17], and an O(log n log log n)-approximation for MIN 2CNF DELETION [14].We also show that the integrality gap of an SDP relaxation of the MINIMUM MULTICUT problem is Ω(log n).
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paper_title: Some optimal inapproximability results

paper_content:

We prove optimal, up to an arbitrary e > 0, inapproximability results for Max-E k-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover.
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paper_title: Economical toric spines via Cheeger's Inequality

paper_content:

Let $G_{\infty}=(C_m^d)_{\infty}$ denote the graph whose set of vertices is $\{1,..., m\}^d$, where two distinct vertices are adjacent iff they are either equal or adjacent in $C_m$ in each coordinate. Let $G_{1}=(C_m^d)_1$ denote the graph on the same set of vertices in which two vertices are adjacent iff they are adjacent in one coordinate in $C_m$ and equal in all others. Both graphs can be viewed as graphs of the $d$-dimensional torus. We prove that one can delete $O(\sqrt d m^{d-1})$ vertices of $G_1$ so that no topologically nontrivial cycles remain. This improves an $O(d^{\log_2 (3/2)}m^{d-1})$ estimate of Bollob\'as, Kindler, Leader and O'Donnell. We also give a short proof of a result implicit in a recent paper of Raz: one can delete an $O(\sqrt d/m)$ fraction of the edges of $G_{\infty}$ so that no topologically nontrivial cycles remain in this graph. Our technique also yields a short proof of a recent result of Kindler, O'Donnell, Rao and Wigderson; there is a subset of the continuous $d$-dimensional torus of surface area $O(\sqrt d)$ that intersects all nontrivial cycles. All proofs are based on the same general idea: the consideration of random shifts of a body with small boundary and no- nontrivial cycles, whose existence is proved by applying the isoperimetric inequality of Cheeger or its vertex or edge discrete analogues.
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paper_title: Optimization, approximation, and complexity classes

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We define a natural variant of NP, MAX NP , and also a subclass called MAX SNP . These are classes of optimization problems, and in fact contain several natural, well-studied ones. We show that problems in these classes can be approximated with some bounded error. Furthermore, we show that a number of common optimization problems are complete under a kind of careful transformation (called L - reduction ) that preserves approximability. It follows that such a complete problem has a polynomial-time approximation scheme iff the whole class does. These results may help explain the lack of progress on the approximability of a host of optimization problems.
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paper_title: Hardness of approximate hypergraph coloring

paper_content:

We introduce the notion of covering complexity of a probabilistic verifier. The covering complexity of a verifier on a given input is the minimum number of proofs needed to "satisfy" the verifier on every random string, i.e., on every random string, at least one of the given proofs must be accepted by the verifier. The covering complexity of PCP verifiers offers a promising route to getting stronger inapproximability results for some minimization problems, and in particular (hyper)-graph coloring problems. We present a PCP verifier for NP statements that queries only four bits and yet has a covering complexity of one for true statements and a super-constant covering complexity for statements not in the language. Moreover the acceptance predicate of this verifier is a simple Not-all-Equal check on the four bits it reads. This enables us to prove that for any constant c, it is NP-hard to color a 2-colorable 4-uniform hypergraph using just c colors, and also yields a super-constant inapproximability result under a stronger hardness assumption.
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paper_title: Some optimal inapproximability results

paper_content:

We prove optimal, up to an arbitrary e > 0, inapproximability results for Max-E k-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover.
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paper_title: Hardness of approximate hypergraph coloring

paper_content:

We introduce the notion of covering complexity of a probabilistic verifier. The covering complexity of a verifier on a given input is the minimum number of proofs needed to "satisfy" the verifier on every random string, i.e., on every random string, at least one of the given proofs must be accepted by the verifier. The covering complexity of PCP verifiers offers a promising route to getting stronger inapproximability results for some minimization problems, and in particular (hyper)-graph coloring problems. We present a PCP verifier for NP statements that queries only four bits and yet has a covering complexity of one for true statements and a super-constant covering complexity for statements not in the language. Moreover the acceptance predicate of this verifier is a simple Not-all-Equal check on the four bits it reads. This enables us to prove that for any constant c, it is NP-hard to color a 2-colorable 4-uniform hypergraph using just c colors, and also yields a super-constant inapproximability result under a stronger hardness assumption.
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paper_title: Some optimal inapproximability results

paper_content:

We prove optimal, up to an arbitrary e > 0, inapproximability results for Max-E k-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover.
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paper_title: On the power of unique 2-prover 1-round games

paper_content:

A 2-prover game is called unique if the answer of one prover uniquely determines the answer of the second prover and vice versa (we implicitly assume games to be one round games). The value of a 2-prover game is the maximum acceptance probability of the verifier over all the prover strategies. We make a conjecture regarding the power of unique 2-prover games, which we call the Unique Games Conjecture.
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paper_title: Inapproximability Results for Sparsest Cut, Optimal Linear Arrangement, and Precedence Constrained Scheduling

paper_content:

We consider (uniform) sparsest cut, optimal linear arrangement and the precedence constrained scheduling problem 1 |prec| SigmawjCj-So far, these three notorious NP-hard problems have resisted all attempts to prove inapproximability results. We show that they have no polynomial time approximation scheme (PTAS), unless NP-complete pmblems can be solved in randomized subexponential time. Furthermore, we prove that the scheduling problem is as-hard to approximate as vertex cover when the so-called fixed cost, that is present in all feasible solutions, is subtracted from the objective function.
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paper_title: Some optimal inapproximability results

paper_content:

We prove optimal, up to an arbitrary e > 0, inapproximability results for Max-E k-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime p and Set Splitting. As a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously. In particular, for Max-E2-Sat, Max-Cut, Max-di-Cut, and Vertex cover.
---
paper_title: Inapproximability of NP-complete problems, discrete fourier analysis, and geometry

paper_content:

This article gives a survey of recent results that connect three areas in com- puter science and mathematics: (1) (Hardness of) computing approximate solutions to NP-complete problems. (2) Fourier analysis of boolean functions on boolean hypercube. (3) Certain problems in geometry, especially related to isoperimetry and embeddings between metric spaces.
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paper_title: On the power of unique 2-prover 1-round games

paper_content:

A 2-prover game is called unique if the answer of one prover uniquely determines the answer of the second prover and vice versa (we implicitly assume games to be one round games). The value of a 2-prover game is the maximum acceptance probability of the verifier over all the prover strategies. We make a conjecture regarding the power of unique 2-prover games, which we call the Unique Games Conjecture.
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paper_title: Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?

paper_content:

In this paper, we give evidence suggesting that MAX-CUT is NP-hard to approximate to within a factor of /spl alpha//sub cw/+ /spl epsi/, for all /spl epsi/ > 0, where /spl alpha//sub cw/ denotes the approximation ratio achieved by the Goemans-Williamson algorithm (1995). /spl alpha//sub cw/ /spl ap/ .878567. This result is conditional, relying on two conjectures: a) the unique games conjecture of Khot; and, b) a very believable conjecture we call the majority is stablest conjecture. These results indicate that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT problem. The same two conjectures also imply that it is NP-hard to (/spl beta/ + /spl epsi/)-approximate MAX-2SAT, where /spl beta/ /spl ap/ .943943 is the minimum of (2 + (2//spl pi/) /spl theta/)/(3 - cos(/spl theta/)) on (/spl pi//2, /spl pi/). Motivated by our proof techniques, we show that if the MAX-2CSP and MAX-2SAT problems are slightly restricted - in a way that seems to retain all their hardness -then they have (/spl alpha//sub GW/-/spl epsi/)- and (/spl beta/ - /spl epsi/)-approximation algorithms, respectively. Though we are unable to prove the majority is stablest conjecture, we give some partial results and indicate possible directions of attack. Our partial results are enough to imply that MAX-CUT is hard to (3/4 + 1/(2/spl pi/) + /spl epsi/)-approximate (/spl ap/ .909155), assuming only the unique games conjecture. We also discuss MAX-2CSP problems over non-Boolean domains and state some related results and conjectures. We show, for example, that the unique games conjecture implies that it is hard to approximate MAX-2LIN(q) to within any constant factor.
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paper_title: How good is the Goemans-Williamson MAX CUT algorithm?

paper_content:

The celebrated semidefinite programming algorithm for MAX CUT introduced by Goemans and Williamson was known to have a performance ratio of at least α = 2 π min0<θ≤π θ 1−cos θ (0.87856 < α < 0.87857); the exact performance ratio was unknown. We prove that the performance ratio of their algorithm is exactly α. Furthermore, we show that it is impossible to add valid linear constraints to improve the performance ratio.
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paper_title: On the integrality ratio of semidefinite relaxations of MAX CUT

paper_content:

MAX CUT is the problem of partitioning the vertices of a graph into two sets, maximizing the number of edges joining these sets. This problem is NP-hard. Goemans and Williamson proposed an algorithm that first uses a semidefinite programming relaxation of MAX CUT to embed the vertices of the graph on the surface of an n dimensional sphere, and then uses a random hyperplane to cut the sphere in two, giving a cut of the graph. They show that the expected number of edges in the random cut is at least α \cdot sdp, where α \simeq 0.87856 and sdp is the value of the semidefinite program. This manuscript shows the following results: 1. The integrality ratio of the semidefinite program is α. The previously known bound on theintegrality ratio was roughly 0.8845. 2. In the presence of the so called “triangle constraints”, the integrality ratio is no better than roughly 0.891. The previously known bound was above 0.95.
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paper_title: Parallel Repetition in Projection Games and a Concentration Bound

paper_content:

A two-player game is played by cooperating players who are not allowed to communicate. A referee asks the players questions sampled from some known distribution and decides whether they win or not based on a known predicate of the questions and the players' answers. The parallel repetition of the game is the game in which the referee samples $n$ independent pairs of questions and sends the corresponding questions to the players simultaneously. If the players cannot win the original game with probability better than $(1-\epsilon)$, what's the best they can do in the repeated game? We improve earlier results of [R. Raz, SIAM J. Comput., 27 (1998), pp. 763-803] and [T. Holenstein, Theory Comput., 5 (2009), pp. 141-172], who showed that the players cannot win all copies in the repeated game with probability better than $(1-\epsilon/2)^{\Omega(n\epsilon^2/c)}$ (here $c$ is the length of the answers in the game), in the following ways: (i) We show that the probability of winning all copies is $(1-\epsilon/2)^{\Omega(\epsilon n)}$ as long as the game is a “projection game,” the type of game most commonly used in hardness of approximation results. (ii) We prove a concentration bound for parallel repetition (of general games) showing that for any constant $0 0$, there exists an alphabet size $M(\epsilon)$ for which it is NP-hard to distinguish a unique game with alphabet size $M$ in which a $(1-\epsilon^2)$ fraction of the constraints can be satisfied from one in which a $(1-\epsilon f(1/\epsilon))$ fraction of the constraints can be satisfied.
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paper_title: Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?

paper_content:

In this paper, we give evidence suggesting that MAX-CUT is NP-hard to approximate to within a factor of /spl alpha//sub cw/+ /spl epsi/, for all /spl epsi/ > 0, where /spl alpha//sub cw/ denotes the approximation ratio achieved by the Goemans-Williamson algorithm (1995). /spl alpha//sub cw/ /spl ap/ .878567. This result is conditional, relying on two conjectures: a) the unique games conjecture of Khot; and, b) a very believable conjecture we call the majority is stablest conjecture. These results indicate that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT problem. The same two conjectures also imply that it is NP-hard to (/spl beta/ + /spl epsi/)-approximate MAX-2SAT, where /spl beta/ /spl ap/ .943943 is the minimum of (2 + (2//spl pi/) /spl theta/)/(3 - cos(/spl theta/)) on (/spl pi//2, /spl pi/). Motivated by our proof techniques, we show that if the MAX-2CSP and MAX-2SAT problems are slightly restricted - in a way that seems to retain all their hardness -then they have (/spl alpha//sub GW/-/spl epsi/)- and (/spl beta/ - /spl epsi/)-approximation algorithms, respectively. Though we are unable to prove the majority is stablest conjecture, we give some partial results and indicate possible directions of attack. Our partial results are enough to imply that MAX-CUT is hard to (3/4 + 1/(2/spl pi/) + /spl epsi/)-approximate (/spl ap/ .909155), assuming only the unique games conjecture. We also discuss MAX-2CSP problems over non-Boolean domains and state some related results and conjectures. We show, for example, that the unique games conjecture implies that it is hard to approximate MAX-2LIN(q) to within any constant factor.
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paper_title: Conditional hardness for approximate coloring

paper_content:

We study the AprxColoring$(q,Q)$ problem: Given a graph $G$, decide whether $\chi(G)\le q$ or $\chi(G)\ge Q$. We present hardness results for this problem for any constants $3\le q<Q$. For $q\ge4$, our result is based on Khot's 2-to-1 label cover, which is conjectured to be NP-hard [S. Khot, Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767-775]. For $q=3$, we base our hardness result on a certain “${\rhd\hskip-0.5em<}$-shaped” variant of his conjecture. Previously no hardness result was known for $q=3$ and $Q\ge6$. At the heart of our proof are tight bounds on generalized noise-stability quantities, which extend the recent work of Mossel, O'Donnell, and Oleszkiewicz [“Noise stability of functions with low influences: Invariance and optimality,” Ann. of Math. (2), to appear] and should have wider applicability.
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paper_title: New approximation guarantee for chromatic number

paper_content:

We describe how to color every 3-colorable graph with O(n0.2111) colors, thus improving an algorithm of Blum and Karger from almost a decade ago. Our analysis uses new geometric ideas inspired by the recent work of Arora, Rao, and Vazirani on SPARSEST CUT, and these ideas show promise of leading to further improvements.
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paper_title: Inapproximability of NP-complete problems, discrete fourier analysis, and geometry

paper_content:

This article gives a survey of recent results that connect three areas in com- puter science and mathematics: (1) (Hardness of) computing approximate solutions to NP-complete problems. (2) Fourier analysis of boolean functions on boolean hypercube. (3) Certain problems in geometry, especially related to isoperimetry and embeddings between metric spaces.
---
paper_title: On the power of unique 2-prover 1-round games

paper_content:

A 2-prover game is called unique if the answer of one prover uniquely determines the answer of the second prover and vice versa (we implicitly assume games to be one round games). The value of a 2-prover game is the maximum acceptance probability of the verifier over all the prover strategies. We make a conjecture regarding the power of unique 2-prover games, which we call the Unique Games Conjecture.
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paper_title: Approximation Resistant Predicates from Pairwise Independence

paper_content:

We study the approximability of predicates on k variables from a domain [q], and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. ...
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paper_title: Gowers uniformity, influence of variables, and PCPs

paper_content:

Gowers introduced, for d\geq 1, the notion of dimension-d uniformity U^d(f) of a function f: G -> \C, where G is a finite abelian group and \C are the complex numbers. Roughly speaking, if U^d(f) is small, then f has certain "pseudorandomness" properties. ::: We prove the following property of functions with large U^d(f). Write G=G_1 x >... x G_n as a product of groups. If a bounded balanced function f:G_1 x ... x G_n -> \C is such that U^{d} (f) > epsilon, then one of the coordinates of f has influence at least epsilon/2^{O(d)}. ::: The Gowers inner product of a collection of functions is a related notion of pseudorandomness. We prove that if a collection of bounded functions has large Gowers inner product, and at least one function in the collection is balanced, then there is a variable that has high influence for at least four of the functions in the collection. ::: Finally, we relate the acceptance probability of the "hypergraph long-code test" proposed by Samorodnitsky and Trevisan to the Gowers inner product of the functions being tested and we deduce applications to the construction of Probabilistically Checkable Proofs and to hardness of approximation.
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paper_title: Optimal algorithms and inapproximability results for every CSP?

paper_content:

Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the following result : If UGC is true, then for every constraint satisfaction problem(CSP) the best approximation ratio is given by a certain simple SDP. Specifically, we show a generic conversion from SDP integrality gaps to UGC hardness results for every CSP. This result holds both for maximization and minimization problems over arbitrary finite domains. Using this connection between integrality gaps and hardness results we obtain a generic polynomial-time algorithm for all CSPs. Assuming the Unique Games Conjecture, this algorithm achieves the optimal approximation ratio for every CSP. Unconditionally, for all 2-CSPs the algorithm achieves an approximation ratio equal to the integrality gap of a natural SDP used in literature. Further the algorithm achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut and Unique Games.
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paper_title: Towards computing the grothendieck constant

paper_content:

The Grothendieck constant KG is the smallest constant such that for every d ∈ N and every matrix A = (aij), ::: ::: [EQUATION] ::: ::: where B(d) is the unit ball in Rd. Despite several efforts [15, 23], the value of the constant KG remains unknown. The Grothendieck constant KG is precisely the integrality gap of a natural SDP relaxation for the KM, N-Quadratic Programming problem. The input to this problem is a matrix A = (aij) and the objective is to maximize the quadratic form Σij aijxiyj over xiyj ∈ [−1, 1]. ::: ::: In this work, we apply techniques from [22] to the KM, N-Quadratic Programming problem. Using some standard but non-trivial modifications, the reduction in [22] yields the following hardness result: Assuming the Unique Games Conjecture [9], it is NP-hard to approximate the KM, N-Quadratic Programming problem to any factor better than the Grothendieck constant KG. ::: ::: By adapting a "bootstrapping" argument used in a proof of Grothendieck inequality [5], we are able to perform a tighter analysis than [22]. Through this careful analysis, we obtain the following new results: ::: ::: • An approximation algorithm for KM, N-Quadratic Programming that is guaranteed to achieve an approximation ratio arbitrarily close to the Grothendieck constant KG (optimal approximation ratio assuming the Unique Games Conjecture). ::: ::: • We show that the Grothendieck constant KG can be computed within an error η, in time depending only on η. Specifically, for each η, we formulate an explicit finite linear program, whose optimum is η-close to the Grothendieck constant. ::: ::: We also exhibit a simple family of operators on the Gaussian Hilbert space that is guaranteed to contain tight examples for the Grothendieck inequality.
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paper_title: Sdp gaps and ugc hardness for multiway cut, 0-extension, and metric labeling

paper_content:

The connection between integrality gaps and computational hardness of discrete optimization problems is an intriguing question. In recent years, this connection has prominently figured in several tight UGC-based hardness results. We show in this paper a direct way of turning integrality gaps into hardness results for several fundamental classification problems. Specifically, we convert linear programming integrality gaps for the Multiway Cut, 0-Extension, and and Metric Labeling problems into UGC-based hardness results. Qualitatively, our result suggests that if the unique games conjecture is true then a linear relaxation of the latter problems studied in several papers (so-called earthmover linear program) yields the best possible approximation. Taking this a step further, we also obtain integrality gaps for a semi-definite programming relaxation matching the integrality gaps of the earthmover linear program. Prior to this work, there was an intriguing possibility of obtaining better approximation factors for labeling problems via semi-definite programming.
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paper_title: Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?

paper_content:

In this paper, we give evidence suggesting that MAX-CUT is NP-hard to approximate to within a factor of /spl alpha//sub cw/+ /spl epsi/, for all /spl epsi/ > 0, where /spl alpha//sub cw/ denotes the approximation ratio achieved by the Goemans-Williamson algorithm (1995). /spl alpha//sub cw/ /spl ap/ .878567. This result is conditional, relying on two conjectures: a) the unique games conjecture of Khot; and, b) a very believable conjecture we call the majority is stablest conjecture. These results indicate that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT problem. The same two conjectures also imply that it is NP-hard to (/spl beta/ + /spl epsi/)-approximate MAX-2SAT, where /spl beta/ /spl ap/ .943943 is the minimum of (2 + (2//spl pi/) /spl theta/)/(3 - cos(/spl theta/)) on (/spl pi//2, /spl pi/). Motivated by our proof techniques, we show that if the MAX-2CSP and MAX-2SAT problems are slightly restricted - in a way that seems to retain all their hardness -then they have (/spl alpha//sub GW/-/spl epsi/)- and (/spl beta/ - /spl epsi/)-approximation algorithms, respectively. Though we are unable to prove the majority is stablest conjecture, we give some partial results and indicate possible directions of attack. Our partial results are enough to imply that MAX-CUT is hard to (3/4 + 1/(2/spl pi/) + /spl epsi/)-approximate (/spl ap/ .909155), assuming only the unique games conjecture. We also discuss MAX-2CSP problems over non-Boolean domains and state some related results and conjectures. We show, for example, that the unique games conjecture implies that it is hard to approximate MAX-2LIN(q) to within any constant factor.
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paper_title: Approximation algorithms for unique games

paper_content:

We present a polynomial time algorithm based on semidefinite programming that, given a unique game of value 1 - O(1/logn), satisfies a constant fraction of constraints, where n is the number of variables. For sufficiently large alphabets, it improves an algorithm of Khot (STOC'02) that satisfies a constant fraction of constraints in unique games of value 1 -O(1/(k/sup 10/(log k)/sup 5/)), where k is the size of the alphabet. We also present a simpler algorithm for the special case of unique games with linear constraints. Finally, we present a simple approximation algorithm for 2-to-1 games.
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paper_title: On the power of unique 2-prover 1-round games

paper_content:

A 2-prover game is called unique if the answer of one prover uniquely determines the answer of the second prover and vice versa (we implicitly assume games to be one round games). The value of a 2-prover game is the maximum acceptance probability of the verifier over all the prover strategies. We make a conjecture regarding the power of unique 2-prover games, which we call the Unique Games Conjecture.
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paper_title: Approximating unique games

paper_content:

The UNIQUE GAMES problem is the following: we are given a graph G = (V, E), with each edge e = (u, v) having a weight w e and a permutation π uv on [k]. The objective is to find a labeling of each vertex u with a label f u ∈ [k] to minimize the weight of unsatisfied edges---where an edge (u, v) is satisfied if f v = π uv (f u ).The Unique Games Conjecture of Khot [8] essentially says that for each e > 0, there is a k such that it is NP-hard to distinguish instances of Unique games with (1-e) satisfiable edges from those with only e satisfiable edges. Several hardness results have recently been proved based on this assumption, including optimal ones for Max-Cut, Vertex-Cover and other problems, making it an important challenge to prove or refute the conjecture.In this paper, we give an O(log n)-approximation algorithm for the problem of minimizing the number of unsatisfied edges in any Unique game. Previous results of Khot [8] and Trevisan [12] imply that if the optimal solution has OPT = em unsatisfied edges, semidefinite relaxations of the problem could give labelings with min {k2e1/5, (e log n)1/2}m unsatisfied edges. In this paper we show how to round a LP relaxation to get an O(log n)-approximation to the problem; i.e., to find a labeling with only O(em log n) = O(OPT log n) unsatisfied edges.
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paper_title: How to Play Unique Games Using Embeddings

paper_content:

In this paper we present a new approximation algorithm for Unique Games. For a Unique Game with n vertices and k states (labels), if a (1 - \varepsilon) fraction of all constraints is satisfiable, the algorithm finds an assignment satisfying a 1 - O\left( {\varepsilon \sqrt {\log n\log k} } \right) fraction of all constraints. To this end, we introduce new embedding techniques for rounding semidefinite relaxations of problems with large domain size.
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paper_title: Approximation algorithms for unique games

paper_content:

We present a polynomial time algorithm based on semidefinite programming that, given a unique game of value 1 - O(1/logn), satisfies a constant fraction of constraints, where n is the number of variables. For sufficiently large alphabets, it improves an algorithm of Khot (STOC'02) that satisfies a constant fraction of constraints in unique games of value 1 -O(1/(k/sup 10/(log k)/sup 5/)), where k is the size of the alphabet. We also present a simpler algorithm for the special case of unique games with linear constraints. Finally, we present a simple approximation algorithm for 2-to-1 games.
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paper_title: Approximating unique games

paper_content:

The UNIQUE GAMES problem is the following: we are given a graph G = (V, E), with each edge e = (u, v) having a weight w e and a permutation π uv on [k]. The objective is to find a labeling of each vertex u with a label f u ∈ [k] to minimize the weight of unsatisfied edges---where an edge (u, v) is satisfied if f v = π uv (f u ).The Unique Games Conjecture of Khot [8] essentially says that for each e > 0, there is a k such that it is NP-hard to distinguish instances of Unique games with (1-e) satisfiable edges from those with only e satisfiable edges. Several hardness results have recently been proved based on this assumption, including optimal ones for Max-Cut, Vertex-Cover and other problems, making it an important challenge to prove or refute the conjecture.In this paper, we give an O(log n)-approximation algorithm for the problem of minimizing the number of unsatisfied edges in any Unique game. Previous results of Khot [8] and Trevisan [12] imply that if the optimal solution has OPT = em unsatisfied edges, semidefinite relaxations of the problem could give labelings with min {k2e1/5, (e log n)1/2}m unsatisfied edges. In this paper we show how to round a LP relaxation to get an O(log n)-approximation to the problem; i.e., to find a labeling with only O(em log n) = O(OPT log n) unsatisfied edges.
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paper_title: How to Play Unique Games Using Embeddings

paper_content:

In this paper we present a new approximation algorithm for Unique Games. For a Unique Game with n vertices and k states (labels), if a (1 - \varepsilon) fraction of all constraints is satisfiable, the algorithm finds an assignment satisfying a 1 - O\left( {\varepsilon \sqrt {\log n\log k} } \right) fraction of all constraints. To this end, we introduce new embedding techniques for rounding semidefinite relaxations of problems with large domain size.
---
paper_title: On the power of unique 2-prover 1-round games

paper_content:

A 2-prover game is called unique if the answer of one prover uniquely determines the answer of the second prover and vice versa (we implicitly assume games to be one round games). The value of a 2-prover game is the maximum acceptance probability of the verifier over all the prover strategies. We make a conjecture regarding the power of unique 2-prover games, which we call the Unique Games Conjecture.
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paper_title: Approximation algorithms for unique games

paper_content:

We present a polynomial time algorithm based on semidefinite programming that, given a unique game of value 1 - O(1/logn), satisfies a constant fraction of constraints, where n is the number of variables. For sufficiently large alphabets, it improves an algorithm of Khot (STOC'02) that satisfies a constant fraction of constraints in unique games of value 1 -O(1/(k/sup 10/(log k)/sup 5/)), where k is the size of the alphabet. We also present a simpler algorithm for the special case of unique games with linear constraints. Finally, we present a simple approximation algorithm for 2-to-1 games.
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paper_title: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms

paper_content:

In this paper, we establish max-flow min-cut theorems for several important classes of multicommodity flow problems. In particular, we show that for any n-node multicommodity flow problem with uniform demands, the max-flow for the problem is within an O(log n) factor of the upper bound implied by the min-cut. The result (which is existentially optimal) establishes an important analogue of the famous 1-commodity max-flow min-cut theorem for problems with multiple commodities. The result also has substantial applications to the field of approximation algorithms. For example, we use the flow result to design the first polynomial-time (polylog n-times-optimal) approximation algorithms for well-known NP-hard optimization problems such as graph partitioning, min-cut linear arrangement, crossing number, VLSI layout, and minimum feedback arc set. Applications of the flow results to path routing problems, network reconfiguration, communication in distributed networks, scientific computing and rapidly mixing Markov chains are also described in the paper. Categories and Subject Descriptors: F.2.2 (Analysis of Algorithms and Problem Complexity):
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paper_title: How to Play Unique Games on Expanders

paper_content:

In this note we improve a recent result by Arora, Khot, Kolla, Steurer, Tulsiani, and Vishnoi on solving the Unique Games problem on expanders. Given a $(1-\varepsilon)$-satisfiable instance of Unique Games with the constraint graph $G$, our algorithm finds an assignment satisfying at least a $1- C \varepsilon/h_G$ fraction of all constraints if $\varepsilon<c \lambda_G$ where $h_G$ is the edge expansion of $G$, $\lambda_G$ is the second smallest eigenvalue of the Laplacian of $G$, and $C$ and $c$ are some absolute constants.
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paper_title: Spectral Algorithms for Unique Games

paper_content:

We present a new algorithm for Unique Games which is based on purely {\em spectral} techniques, in contrast to previous work in the area, which relies heavily on semidefinite programming (SDP). Given a highly satisfiable instance of Unique Games, our algorithm is able to recover a good assignment. The approximation guarantee depends only on the completeness of the game, and not on the alphabet size, while the running time depends on spectral properties of the {\em Label-Extended} graph associated with the instance of Unique Games. In particular, we show how our techniques imply a quasi-polynomial time algorithm that decides satisfiability of a game on the Khot-Vishnoi\cite{KV} integrality gap instance. Notably, when run on that instance, the standard SDP relaxation of Unique Games {\em fails}. As a special case, we also show how to re-derive a polynomial time algorithm for Unique Games on expander constraint graphs (similar to \cite{AKKTSV}) and a sub-exponential time algorithm for Unique Games on the Hypercube.
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paper_title: Graph expansion and the unique games conjecture

paper_content:

The edge expansion of a subset of vertices S ⊆ V in a graph G measures the fraction of edges that leave S. In a d-regular graph, the edge expansion/conductance Φ(S) of a subset S ⊆ V is defined as Φ(S) = (|E(S, V\S)|)/(d|S|). Approximating the conductance of small linear sized sets (size δ n) is a natural optimization question that is a variant of the well-studied Sparsest Cut problem. However, there are no known algorithms to even distinguish between almost complete edge expansion (Φ(S) = 1-ε), and close to 0 expansion. In this work, we investigate the connection between Graph Expansion and the Unique Games Conjecture. Specifically, we show the following: We show that a simple decision version of the problem of approximating small set expansion reduces to Unique Games. Thus if approximating edge expansion of small sets is hard, then Unique Games is hard. Alternatively, a refutation of the UGC will yield better algorithms to approximate edge expansion in graphs. This is the first non-trivial "reverse" reduction from a natural optimization problem to Unique Games. Under a slightly stronger UGC that assumes mild expansion of small sets, we show that it is UG-hard to approximate small set expansion. On instances with sufficiently good expansion of small sets, we show that Unique Games is easy by extending the techniques of [4].
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paper_title: Graph expansion and the unique games conjecture

paper_content:

The edge expansion of a subset of vertices S ⊆ V in a graph G measures the fraction of edges that leave S. In a d-regular graph, the edge expansion/conductance Φ(S) of a subset S ⊆ V is defined as Φ(S) = (|E(S, V\S)|)/(d|S|). Approximating the conductance of small linear sized sets (size δ n) is a natural optimization question that is a variant of the well-studied Sparsest Cut problem. However, there are no known algorithms to even distinguish between almost complete edge expansion (Φ(S) = 1-ε), and close to 0 expansion. In this work, we investigate the connection between Graph Expansion and the Unique Games Conjecture. Specifically, we show the following: We show that a simple decision version of the problem of approximating small set expansion reduces to Unique Games. Thus if approximating edge expansion of small sets is hard, then Unique Games is hard. Alternatively, a refutation of the UGC will yield better algorithms to approximate edge expansion in graphs. This is the first non-trivial "reverse" reduction from a natural optimization problem to Unique Games. Under a slightly stronger UGC that assumes mild expansion of small sets, we show that it is UG-hard to approximate small set expansion. On instances with sufficiently good expansion of small sets, we show that Unique Games is easy by extending the techniques of [4].
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paper_title: Inapproximability of NP-complete problems, discrete fourier analysis, and geometry

paper_content:

This article gives a survey of recent results that connect three areas in com- puter science and mathematics: (1) (Hardness of) computing approximate solutions to NP-complete problems. (2) Fourier analysis of boolean functions on boolean hypercube. (3) Certain problems in geometry, especially related to isoperimetry and embeddings between metric spaces.
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paper_title: Proof verification and the hardness of approximation problems

paper_content:

We show that every language in NP has a probablistic verifier that checks membership proofs for it using logarithmic number of random bits and by examining a constant number of bits in the proof. If a string is in the language, then there exists a proof such that the verifier accepts with probability 1 (i.e., for every choice of its random string). For strings not in the language, the verifier rejects every provided “proof” with probability at least 1/2. Our result builds upon and improves a recent result of Arora and Safra [1998] whose verifiers examine a nonconstant number of bits in the proof (though this number is a very slowly growing function of the input length). ::: As a consequence, we prove that no MAX SNP-hard problem has a polynomial time approximation scheme, unless NP = P. The class MAX SNP was defined by Papadimitriou and Yannakakis [1991] and hard problems for this class include vertex cover, maximum satisfiability, maximum cut, metric TSP, Steiner trees and shortest superstring. We also improve upon the clique hardness results of Feige et al. [1996] and Arora and Safra [1998] and show that there exists a positive ε such that approximating the maximum clique size in an N-vertex graph to within a factor of Nε is NP-hard.
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paper_title: On the power of unique 2-prover 1-round games

paper_content:

A 2-prover game is called unique if the answer of one prover uniquely determines the answer of the second prover and vice versa (we implicitly assume games to be one round games). The value of a 2-prover game is the maximum acceptance probability of the verifier over all the prover strategies. We make a conjecture regarding the power of unique 2-prover games, which we call the Unique Games Conjecture.
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paper_title: An Approximate Zero-One Law

paper_content:

We prove an approximate zero-one law, which holds for finite Bernoulli schemes. An application to percolation theory is given.
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paper_title: Finite metric spaces: combinatorics, geometry and algorithms

paper_content:

In the last several years a number of very interesting results were proved about finite metric spaces. Some of this work is motivated by practical considerations: Large data sets (coming e.g. from computational molecular biology, brain research or data mining) can be viewed as large metric spaces that should be analyzed (e.g. correctly clustered).On the other hand, these investigations connect to some classical areas of geometry - the asymptotic theory of finite-dimensional normed spaces and differential geometry. Finally, the metric theory of finite graphs has proved very useful in the study of graphs per se and the design of approximation algorithms for hard computational problems. In this talk I will try to explain some of the results and review some of the emerging new connections and the many fascinating open problems in this area.
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paper_title: Semidefinite Programming in Combinatorial Optimization

paper_content:

We discuss the use of semidefinite programming for combinatorial optimization problems. The main topics covered include (i) the Lovasz theta function and its applications to stable sets, perfect graphs, and coding theory, (ii) the automatic generation of strong valid inequalities, (iii) the maximum cut problem and related problems, and (iv) the embedding of finite metric spaces and its relationship to the sparsest cut problem.
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paper_title: On the power of unique 2-prover 1-round games

paper_content:

A 2-prover game is called unique if the answer of one prover uniquely determines the answer of the second prover and vice versa (we implicitly assume games to be one round games). The value of a 2-prover game is the maximum acceptance probability of the verifier over all the prover strategies. We make a conjecture regarding the power of unique 2-prover games, which we call the Unique Games Conjecture.
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paper_title: The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into ℓ1

paper_content:

In this paper, we disprove a conjecture of Goemans and Linial; namely, that every negative type metric embeds into $\ell_1$ with constant distortion. We show that for an arbitrarily small constant $\delta> 0$, for all large enough $n$, there is an $n$-point negative type metric which requires distortion at least $(\log\log n)^{1/6-\delta}$ to embed into $\ell_1.$ ::: Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot, establishing a previously unsuspected connection between probabilistically checkable proof systems (PCPs) and the theory of metric embeddings. We first prove that the UGC implies a super-constant hardness result for the (non-uniform) Sparsest Cut problem. Though this hardness result relies on the UGC, we demonstrate, nevertheless, that the corresponding PCP reduction can be used to construct an "integrality gap instance" for Sparsest Cut. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of Unique Games. Then we "simulate" the PCP reduction and "translate" the integrality gap instance of Unique Games to an integrality gap instance of Sparsest Cut. This enables us to prove a $(\log \log n)^{1/6-\delta}$ integrality gap for Sparsest Cut, which is known to be equivalent to the metric embedding lower bound.
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paper_title: Improved lower bounds for embeddings into L1

paper_content:

We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into $L_1$. In particular, we show that for every $n\ge1$, there is an $n$-point metric space of negative type that requires a distortion of $\Omega(\log\log n)$ for such an embedding, implying the same lower bound on the integrality gap of a well-known semidefinite programming relaxation for sparsest cut. This result builds upon and improves the recent lower bound of $(\log\log n)^{1/6-o(1)}$ due to Khot and Vishnoi [The unique games conjecture, integrality gap for cut problems and the embeddability of negative type metrics into $l_1$, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 2005, pp. 53-62]. We also show that embedding the edit distance metric on $\{0,1\}^n$ into $L_1$ requires a distortion of $\Omega(\log n)$. This result improves a very recent $(\log n)^{1/2-o(1)}$ lower bound by Khot and Naor [Nonembeddability theorems via Fourier analysis, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 2005, pp. 101-112].
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paper_title: ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT

paper_content:

We show that the Multicut, Sparsest-Cut, and Min-2CNF ? Deletion problems are NP-hard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot (2002). A quantitatively stronger version of the conjecture implies an inapproximability factor of $$\Omega(\sqrt{\log \log n}).$$
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paper_title: Integrality gaps for sparsest cut and minimum linear arrangement problems

paper_content:

Arora, Rao and Vazirani [2] showed that the standard semi-definite programming (SDP) relaxation of the Sparsest Cut problem with the triangle inequality constraints has an integrality gap of O(√log n). They conjectured that the gap is bounded from above by a constant. In this paper, we disprove this conjecture (referred to as the ARV-Conjecture) by constructing an Ω(log log n) integrality gap instance. Khot and Vishnoi [16] had earlier disproved the non-uniform version of the ARV-Conjecture.A simple "stretching" of the integrality gap instance for the Sparsest Cut problem serves as an Ω(log log n) integrality gap instance for the SDP relaxation of the Minimum Linear Arrangement problem. This SDP relaxation was considered in [6, 11], where it was shown that its integrality gap is bounded from above by O(√log n log log n).
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paper_title: The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into ℓ1

paper_content:

In this paper, we disprove a conjecture of Goemans and Linial; namely, that every negative type metric embeds into $\ell_1$ with constant distortion. We show that for an arbitrarily small constant $\delta> 0$, for all large enough $n$, there is an $n$-point negative type metric which requires distortion at least $(\log\log n)^{1/6-\delta}$ to embed into $\ell_1.$ ::: Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot, establishing a previously unsuspected connection between probabilistically checkable proof systems (PCPs) and the theory of metric embeddings. We first prove that the UGC implies a super-constant hardness result for the (non-uniform) Sparsest Cut problem. Though this hardness result relies on the UGC, we demonstrate, nevertheless, that the corresponding PCP reduction can be used to construct an "integrality gap instance" for Sparsest Cut. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of Unique Games. Then we "simulate" the PCP reduction and "translate" the integrality gap instance of Unique Games to an integrality gap instance of Sparsest Cut. This enables us to prove a $(\log \log n)^{1/6-\delta}$ integrality gap for Sparsest Cut, which is known to be equivalent to the metric embedding lower bound.
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paper_title: Limit theorems for polylinear forms

paper_content:

The limit theorems for polylinear forms are obtained. Conditions are found under which the distribution of the polylinear form of many random variables is essentially the same as if all the distributions of arguments were normal.
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paper_title: Geometric bounds on the Ornstein-Uhlenbeck velocity process

paper_content:

Let X: Ω→C(ℝ+;ℝ n ) be the Ornstein-Uhlenbeck velocity process in equilibrium and denote by τ A =τ A (X) the first hitting time of \(A \subseteq \mathbb{R}^n \). If A, B∈ℛn and ℙ(X(O)∈A=ℙ(X n (O)≦a), ℙ(X n (O)∈B=ℙ(X n (O)≧b)we prove that \(\mathbb{P}(\tau _A \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } t)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } \mathbb{P}(\tau _{\{ \chi _n \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } a\} } \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } t)\) and \(\mathbb{E}\left( {\int\limits_0^{t \wedge \tau A} {1_{\text{B}} (X({\text{s}})d{\text{s}}} } \right)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \mathbb{E}\left( {\int\limits_0^{t \wedge \tau _{\left\{ {x_n \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } a} \right\}} } {1_{\left\{ {x_n \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } b} \right\}} (X({\text{s))}}d{\text{s}}} } \right)\). Here X n denotes the n-th component of X.
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paper_title: Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?

paper_content:

In this paper, we give evidence suggesting that MAX-CUT is NP-hard to approximate to within a factor of /spl alpha//sub cw/+ /spl epsi/, for all /spl epsi/ > 0, where /spl alpha//sub cw/ denotes the approximation ratio achieved by the Goemans-Williamson algorithm (1995). /spl alpha//sub cw/ /spl ap/ .878567. This result is conditional, relying on two conjectures: a) the unique games conjecture of Khot; and, b) a very believable conjecture we call the majority is stablest conjecture. These results indicate that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT problem. The same two conjectures also imply that it is NP-hard to (/spl beta/ + /spl epsi/)-approximate MAX-2SAT, where /spl beta/ /spl ap/ .943943 is the minimum of (2 + (2//spl pi/) /spl theta/)/(3 - cos(/spl theta/)) on (/spl pi//2, /spl pi/). Motivated by our proof techniques, we show that if the MAX-2CSP and MAX-2SAT problems are slightly restricted - in a way that seems to retain all their hardness -then they have (/spl alpha//sub GW/-/spl epsi/)- and (/spl beta/ - /spl epsi/)-approximation algorithms, respectively. Though we are unable to prove the majority is stablest conjecture, we give some partial results and indicate possible directions of attack. Our partial results are enough to imply that MAX-CUT is hard to (3/4 + 1/(2/spl pi/) + /spl epsi/)-approximate (/spl ap/ .909155), assuming only the unique games conjecture. We also discuss MAX-2CSP problems over non-Boolean domains and state some related results and conjectures. We show, for example, that the unique games conjecture implies that it is hard to approximate MAX-2LIN(q) to within any constant factor.
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paper_title: Noise stability of functions with low influences: Invariance and optimality

paper_content:

In this paper we study functions with low influences on product probability spaces. The analysis of boolean functions with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics. ::: We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known non-linear invariance principles. It has the advantage that its proof is simple and that the error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly ``smoothed''; this extension is essential for our applications to ``noise stability''-type problems. ::: In particular, as applications of the invariance principle we prove two conjectures: the ``Majority Is Stablest'' conjecture from theoretical computer science, which was the original motivation for this work, and the ``It Ain't Over Till It's Over'' conjecture from social choice theory.
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paper_title: Optimal Long Code Test with One Free Bit

paper_content:

For arbitrarily small constants epsilon, delta ≫ 0$, we present a long code test with one free bit, completeness 1-epsilon and soundness delta. Using the test, we prove the following two inapproximability results:1. Assuming the Unique Games Conjecture of Khot, given an n-vertex graph that has two disjoint independent sets of size (1/2-epsilon)n each, it is NP-hard to find an independent set of size delta n.2. Assuming a (new) stronger version of the Unique Games Conjecture, the scheduling problem of minimizing weighted completion time with precedence constraints is inapproximable within factor 2-epsilon.
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paper_title: Noise stability of functions with low influences: Invariance and optimality

paper_content:

In this paper we study functions with low influences on product probability spaces. The analysis of boolean functions with low influences has become a central problem in discrete Fourier analysis. It is motivated by fundamental questions arising from the construction of probabilistically checkable proofs in theoretical computer science and from problems in the theory of social choice in economics. ::: We prove an invariance principle for multilinear polynomials with low influences and bounded degree; it shows that under mild conditions the distribution of such polynomials is essentially invariant for all product spaces. Ours is one of the very few known non-linear invariance principles. It has the advantage that its proof is simple and that the error bounds are explicit. We also show that the assumption of bounded degree can be eliminated if the polynomials are slightly ``smoothed''; this extension is essential for our applications to ``noise stability''-type problems. ::: In particular, as applications of the invariance principle we prove two conjectures: the ``Majority Is Stablest'' conjecture from theoretical computer science, which was the original motivation for this work, and the ``It Ain't Over Till It's Over'' conjecture from social choice theory.
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paper_title: Gowers uniformity, influence of variables, and PCPs

paper_content:

Gowers introduced, for d\geq 1, the notion of dimension-d uniformity U^d(f) of a function f: G -> \C, where G is a finite abelian group and \C are the complex numbers. Roughly speaking, if U^d(f) is small, then f has certain "pseudorandomness" properties. ::: We prove the following property of functions with large U^d(f). Write G=G_1 x >... x G_n as a product of groups. If a bounded balanced function f:G_1 x ... x G_n -> \C is such that U^{d} (f) > epsilon, then one of the coordinates of f has influence at least epsilon/2^{O(d)}. ::: The Gowers inner product of a collection of functions is a related notion of pseudorandomness. We prove that if a collection of bounded functions has large Gowers inner product, and at least one function in the collection is balanced, then there is a variable that has high influence for at least four of the functions in the collection. ::: Finally, we relate the acceptance probability of the "hypergraph long-code test" proposed by Samorodnitsky and Trevisan to the Gowers inner product of the functions being tested and we deduce applications to the construction of Probabilistically Checkable Proofs and to hardness of approximation.
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paper_title: Expander flows, geometric embeddings and graph partitioning

paper_content:

We give a O(slog n)-approximation algorithm for the sparsest cut, edge expansion, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in Rd, whose proof makes essential use of a phenomenon called measure concentration. We also describe an interesting and natural “approximate certificate” for a graph's expansion, which involves embedding an n-node expander in it with appropriate dilation and congestion. We call this an expander flow.
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paper_title: Sharp kernel clustering algorithms and their associated Grothendieck inequalities

paper_content:

In the kernel clustering problem we are given a (large) $n\times n$ symmetric positive semidefinite matrix $A=(a_{ij})$ with $\sum_{i=1}^n\sum_{j=1}^n a_{ij}=0$ and a (small) $k\times k$ symmetric positive semidefinite matrix $B=(b_{ij})$. The goal is to find a partition $\{S_1,...,S_k\}$ of $\{1,... n\}$ which maximizes $ \sum_{i=1}^k\sum_{j=1}^k (\sum_{(p,q)\in S_i\times S_j}a_{pq})b_{ij}$. We design a polynomial time approximation algorithm that achieves an approximation ratio of $\frac{R(B)^2}{C(B)}$, where $R(B)$ and $C(B)$ are geometric parameters that depend only on the matrix $B$, defined as follows: if $b_{ij} =$ is the Gram matrix representation of $B$ for some $v_1,...,v_k\in \R^k$ then $R(B)$ is the minimum radius of a Euclidean ball containing the points $\{v_1, ..., v_k\}$. The parameter $C(B)$ is defined as the maximum over all measurable partitions $\{A_1,...,A_k\}$ of $\R^{k-1}$ of the quantity $\sum_{i=1}^k\sum_{j=1}^k b_{ij}$, where for $i\in \{1,...,k\}$ the vector $z_i\in \R^{k-1}$ is the Gaussian moment of $A_i$, i.e., $z_i=\frac{1}{(2\pi)^{(k-1)/2}}\int_{A_i}xe^{-\|x\|_2^2/2}dx$. We also show that for every $\eps>0$, achieving an approximation guarantee of $(1-\e)\frac{R(B)^2}{C(B)}$ is Unique Games hard.
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paper_title: Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?

paper_content:

In this paper, we give evidence suggesting that MAX-CUT is NP-hard to approximate to within a factor of /spl alpha//sub cw/+ /spl epsi/, for all /spl epsi/ > 0, where /spl alpha//sub cw/ denotes the approximation ratio achieved by the Goemans-Williamson algorithm (1995). /spl alpha//sub cw/ /spl ap/ .878567. This result is conditional, relying on two conjectures: a) the unique games conjecture of Khot; and, b) a very believable conjecture we call the majority is stablest conjecture. These results indicate that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT problem. The same two conjectures also imply that it is NP-hard to (/spl beta/ + /spl epsi/)-approximate MAX-2SAT, where /spl beta/ /spl ap/ .943943 is the minimum of (2 + (2//spl pi/) /spl theta/)/(3 - cos(/spl theta/)) on (/spl pi//2, /spl pi/). Motivated by our proof techniques, we show that if the MAX-2CSP and MAX-2SAT problems are slightly restricted - in a way that seems to retain all their hardness -then they have (/spl alpha//sub GW/-/spl epsi/)- and (/spl beta/ - /spl epsi/)-approximation algorithms, respectively. Though we are unable to prove the majority is stablest conjecture, we give some partial results and indicate possible directions of attack. Our partial results are enough to imply that MAX-CUT is hard to (3/4 + 1/(2/spl pi/) + /spl epsi/)-approximate (/spl ap/ .909155), assuming only the unique games conjecture. We also discuss MAX-2CSP problems over non-Boolean domains and state some related results and conjectures. We show, for example, that the unique games conjecture implies that it is hard to approximate MAX-2LIN(q) to within any constant factor.
---
paper_title: Parallel repetition: simplifications and the no-signaling case

paper_content:

Consider a game where a refereed chooses (x,y) according to a publiclyknown distribution PXY, sends x to Alice, and y to Bob. Withoutcommunicating with each other, Alice responds with a value "a" and Bobresponds with a value "b". Alice and Bob jointly win if a publiclyknown predicate Q(x,y,a,b) holds. Let such a game be given and assume that the maximum probabilitythat Alice and Bob can win is v
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paper_title: Probabilistic checking of proofs: a new characterization of NP

paper_content:

We give a new characterization of NP: the class NP contains exactly those languages <italic>L</italic> for which membership proofs (a proof that an input <italic>x</italic> is in <italic>L</italic>) can be verified probabilistically in polynomial time using <italic>logarithmic</italic> number of random bits and by reading <italic>sublogarithmic</italic> number of bits from the proof. ::: We discuss implications of this characterization; specifically, we show that approximating Clique and Independent Set, even in a very weak sense, is NP-hard.
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paper_title: Parallel Repetition in Projection Games and a Concentration Bound

paper_content:

A two-player game is played by cooperating players who are not allowed to communicate. A referee asks the players questions sampled from some known distribution and decides whether they win or not based on a known predicate of the questions and the players' answers. The parallel repetition of the game is the game in which the referee samples $n$ independent pairs of questions and sends the corresponding questions to the players simultaneously. If the players cannot win the original game with probability better than $(1-\epsilon)$, what's the best they can do in the repeated game? We improve earlier results of [R. Raz, SIAM J. Comput., 27 (1998), pp. 763-803] and [T. Holenstein, Theory Comput., 5 (2009), pp. 141-172], who showed that the players cannot win all copies in the repeated game with probability better than $(1-\epsilon/2)^{\Omega(n\epsilon^2/c)}$ (here $c$ is the length of the answers in the game), in the following ways: (i) We show that the probability of winning all copies is $(1-\epsilon/2)^{\Omega(\epsilon n)}$ as long as the game is a “projection game,” the type of game most commonly used in hardness of approximation results. (ii) We prove a concentration bound for parallel repetition (of general games) showing that for any constant $0 0$, there exists an alphabet size $M(\epsilon)$ for which it is NP-hard to distinguish a unique game with alphabet size $M$ in which a $(1-\epsilon^2)$ fraction of the constraints can be satisfied from one in which a $(1-\epsilon f(1/\epsilon))$ fraction of the constraints can be satisfied.
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paper_title: Approximation Resistant Predicates from Pairwise Independence

paper_content:

We study the approximability of predicates on k variables from a domain [q], and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. ...
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paper_title: Understanding Parallel Repetition Requires Understanding Foams

paper_content:

Motivated by the study of parallel repetition and also by the unique games conjecture, we investigate the value of the "odd cycle games" under parallel repetition. Using tools from discrete harmonic analysis, we show that after d rounds on the cycle of length m, the value of the game is at most 1-(1/m)ldrOmega macr(radicd) (for dlesm2, say). This beats the natural barrier of 1-Theta(1/m)2 ldrd for Raz-style proofs and also the SDP bound of Feige-Lovasz; however, it just barely fails to have implications for unique games. On the other hand, we also show that improving our bound would require proving nontrivial lower bounds on the surface area of high-dimensional foams. Specifically, one would need to answer: what is the least surface area of a cell that tiles Rd by the lattice Zd?
---
paper_title: A Counterexample to Strong Parallel Repetition

paper_content:

The parallel repetition theorem states that, for any two-prover game with value $1-\epsilon$ (for, say, $\epsilon\leq1/2$), the value of the game repeated in parallel $n$ times is at most $(1-\epsilon^c)^{\Omega(n/s)}$, where $s$ is the answer's length (of the original game) and $c$ is a universal constant [R. Raz, SIAM J. Comput., 27 (1998), pp. 763-803]. Several researchers asked whether this bound could be improved to $(1-\epsilon)^{\Omega(n/s)}$; this question is usually referred to as the strong parallel repetition problem. We show that the answer to this question is negative. More precisely, we consider the odd cycle game of size $m$, a two-prover game with value $1-1/2m$. We show that the value of the odd cycle game repeated in parallel $n$ times is at least $1-(1/m)\cdot O(\sqrt{n})$. This implies that, for large enough $n$ (say, $n\geq\Omega(m^2)$), the value of the odd cycle game repeated in parallel $n$ times is at least $(1-1/4m^2)^{O(n)}$. Thus the following hold. 1. For parallel repetition of general games, the bounds of $(1-\epsilon^c)^{\Omega(n/s)}$ given in [R. Raz, SIAM J. Comput., 27 (1998), pp. 763-803; T. Holenstein, in Proceedings of STOC 2002, ACM, New York, 2002, pp. 767-775] are of the right form, up to determining the exact value of the constant $c\geq2$. 2. For parallel repetition of XOR games, unique games, and projection games, the bounds of $(1-\epsilon^2)^{\Omega(n)}$ given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179-192] (for XOR games) and in [A. Rao, in Proceedings of STOC 2008, ACM, New York, 2008, pp. 1-10] (for unique and projection games) are tight. 3. For parallel repetition of the odd cycle game, the bound of $1-(1/m)\cdot\tilde{\Omega}(\sqrt{n})$ given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179-192] is almost tight. A major motivation for the recent interest in the strong parallel repetition problem is that a strong parallel repetition theorem would have implied that the unique game conjecture is equivalent to the NP hardness of distinguishing between instances of Max-Cut that are at least $1-\epsilon^2$ satisfiable from instances that are at most $1-(2/\pi)\cdot\epsilon$ satisfiable. Our results suggest that this cannot be proved just by improving the known bounds on parallel repetition.
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paper_title: Consequences and limits of nonlocal strategies

paper_content:

This paper investigates the powers and limitations of quantum entanglement in the context of cooperative games of incomplete information. We give several examples of such nonlocal games where strategies that make use of entanglement outperform all possible classical strategies. One implication of these examples is that entanglement can profoundly affect the soundness property of two-prover interactive proof systems. We then establish limits on the probability with which strategies making use of entanglement can win restricted types of nonlocal games. These upper bounds may be regarded as generalizations of Tsirelson-type inequalities, which place bounds on the extent to which quantum information can allow for the violation of Bell inequalities. We also investigate the amount of entanglement required by optimal and nearly optimal quantum strategies for some games.
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paper_title: Rounding Parallel Repetitions of Unique Games

paper_content:

We show a connection between the semidefinite relaxation of unique games and their behavior under parallel repetition. Specifically,denoting by val(G) the value of a two-prover unique game G, andby sdpval(G) the value of a natural semidefinite program to approximate val(G), we prove that for every l epsi N, if sdpval(G) ges 1-delta, then val(Gl) ges 1-radicsldelta. Here, Gl denotes the l-fold parallel repetition of G, and s=O(log(k/delta)), where k denotes the alphabet size of the game. For the special case where G is an XOR game (i.e., k=2), we obtain the same bound but with s as an absolute constant. Our bounds on s are optimal up to a factor of O(log(1/delta)). For games with a significant gap between the quantities val(G) and sdpval(G), our result implies that val(Gl) may be much larger than val(G)l, giving a counterexample to the strong parallel repetition conjecture. In a recent breakthrough, Raz (FOCS'08) has shown such an example using the max-cut game on oddcycles. Our results are based on a generalization of his techniques.
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paper_title: Inapproximability Results for Sparsest Cut, Optimal Linear Arrangement, and Precedence Constrained Scheduling

paper_content:

We consider (uniform) sparsest cut, optimal linear arrangement and the precedence constrained scheduling problem 1 |prec| SigmawjCj-So far, these three notorious NP-hard problems have resisted all attempts to prove inapproximability results. We show that they have no polynomial time approximation scheme (PTAS), unless NP-complete pmblems can be solved in randomized subexponential time. Furthermore, we prove that the scheduling problem is as-hard to approximate as vertex cover when the so-called fixed cost, that is present in all feasible solutions, is subtracted from the objective function.
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paper_title: Sharp kernel clustering algorithms and their associated Grothendieck inequalities

paper_content:

In the kernel clustering problem we are given a (large) $n\times n$ symmetric positive semidefinite matrix $A=(a_{ij})$ with $\sum_{i=1}^n\sum_{j=1}^n a_{ij}=0$ and a (small) $k\times k$ symmetric positive semidefinite matrix $B=(b_{ij})$. The goal is to find a partition $\{S_1,...,S_k\}$ of $\{1,... n\}$ which maximizes $ \sum_{i=1}^k\sum_{j=1}^k (\sum_{(p,q)\in S_i\times S_j}a_{pq})b_{ij}$. We design a polynomial time approximation algorithm that achieves an approximation ratio of $\frac{R(B)^2}{C(B)}$, where $R(B)$ and $C(B)$ are geometric parameters that depend only on the matrix $B$, defined as follows: if $b_{ij} =$ is the Gram matrix representation of $B$ for some $v_1,...,v_k\in \R^k$ then $R(B)$ is the minimum radius of a Euclidean ball containing the points $\{v_1, ..., v_k\}$. The parameter $C(B)$ is defined as the maximum over all measurable partitions $\{A_1,...,A_k\}$ of $\R^{k-1}$ of the quantity $\sum_{i=1}^k\sum_{j=1}^k b_{ij}$, where for $i\in \{1,...,k\}$ the vector $z_i\in \R^{k-1}$ is the Gaussian moment of $A_i$, i.e., $z_i=\frac{1}{(2\pi)^{(k-1)/2}}\int_{A_i}xe^{-\|x\|_2^2/2}dx$. We also show that for every $\eps>0$, achieving an approximation guarantee of $(1-\e)\frac{R(B)^2}{C(B)}$ is Unique Games hard.
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paper_title: The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into ℓ1

paper_content:

In this paper, we disprove a conjecture of Goemans and Linial; namely, that every negative type metric embeds into $\ell_1$ with constant distortion. We show that for an arbitrarily small constant $\delta> 0$, for all large enough $n$, there is an $n$-point negative type metric which requires distortion at least $(\log\log n)^{1/6-\delta}$ to embed into $\ell_1.$ ::: Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot, establishing a previously unsuspected connection between probabilistically checkable proof systems (PCPs) and the theory of metric embeddings. We first prove that the UGC implies a super-constant hardness result for the (non-uniform) Sparsest Cut problem. Though this hardness result relies on the UGC, we demonstrate, nevertheless, that the corresponding PCP reduction can be used to construct an "integrality gap instance" for Sparsest Cut. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of Unique Games. Then we "simulate" the PCP reduction and "translate" the integrality gap instance of Unique Games to an integrality gap instance of Sparsest Cut. This enables us to prove a $(\log \log n)^{1/6-\delta}$ integrality gap for Sparsest Cut, which is known to be equivalent to the metric embedding lower bound.
---
paper_title: Optimal inapproximability results for MAX-CUT and other 2-variable CSPs?

paper_content:

In this paper, we give evidence suggesting that MAX-CUT is NP-hard to approximate to within a factor of /spl alpha//sub cw/+ /spl epsi/, for all /spl epsi/ > 0, where /spl alpha//sub cw/ denotes the approximation ratio achieved by the Goemans-Williamson algorithm (1995). /spl alpha//sub cw/ /spl ap/ .878567. This result is conditional, relying on two conjectures: a) the unique games conjecture of Khot; and, b) a very believable conjecture we call the majority is stablest conjecture. These results indicate that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT problem. The same two conjectures also imply that it is NP-hard to (/spl beta/ + /spl epsi/)-approximate MAX-2SAT, where /spl beta/ /spl ap/ .943943 is the minimum of (2 + (2//spl pi/) /spl theta/)/(3 - cos(/spl theta/)) on (/spl pi//2, /spl pi/). Motivated by our proof techniques, we show that if the MAX-2CSP and MAX-2SAT problems are slightly restricted - in a way that seems to retain all their hardness -then they have (/spl alpha//sub GW/-/spl epsi/)- and (/spl beta/ - /spl epsi/)-approximation algorithms, respectively. Though we are unable to prove the majority is stablest conjecture, we give some partial results and indicate possible directions of attack. Our partial results are enough to imply that MAX-CUT is hard to (3/4 + 1/(2/spl pi/) + /spl epsi/)-approximate (/spl ap/ .909155), assuming only the unique games conjecture. We also discuss MAX-2CSP problems over non-Boolean domains and state some related results and conjectures. We show, for example, that the unique games conjecture implies that it is hard to approximate MAX-2LIN(q) to within any constant factor.
---
paper_title: On the integrality ratio of semidefinite relaxations of MAX CUT

paper_content:

MAX CUT is the problem of partitioning the vertices of a graph into two sets, maximizing the number of edges joining these sets. This problem is NP-hard. Goemans and Williamson proposed an algorithm that first uses a semidefinite programming relaxation of MAX CUT to embed the vertices of the graph on the surface of an n dimensional sphere, and then uses a random hyperplane to cut the sphere in two, giving a cut of the graph. They show that the expected number of edges in the random cut is at least α \cdot sdp, where α \simeq 0.87856 and sdp is the value of the semidefinite program. This manuscript shows the following results: 1. The integrality ratio of the semidefinite program is α. The previously known bound on theintegrality ratio was roughly 0.8845. 2. In the presence of the so called “triangle constraints”, the integrality ratio is no better than roughly 0.891. The previously known bound was above 0.95.
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paper_title: Graph expansion and the unique games conjecture

paper_content:

The edge expansion of a subset of vertices S ⊆ V in a graph G measures the fraction of edges that leave S. In a d-regular graph, the edge expansion/conductance Φ(S) of a subset S ⊆ V is defined as Φ(S) = (|E(S, V\S)|)/(d|S|). Approximating the conductance of small linear sized sets (size δ n) is a natural optimization question that is a variant of the well-studied Sparsest Cut problem. However, there are no known algorithms to even distinguish between almost complete edge expansion (Φ(S) = 1-ε), and close to 0 expansion. In this work, we investigate the connection between Graph Expansion and the Unique Games Conjecture. Specifically, we show the following: We show that a simple decision version of the problem of approximating small set expansion reduces to Unique Games. Thus if approximating edge expansion of small sets is hard, then Unique Games is hard. Alternatively, a refutation of the UGC will yield better algorithms to approximate edge expansion in graphs. This is the first non-trivial "reverse" reduction from a natural optimization problem to Unique Games. Under a slightly stronger UGC that assumes mild expansion of small sets, we show that it is UG-hard to approximate small set expansion. On instances with sufficiently good expansion of small sets, we show that Unique Games is easy by extending the techniques of [4].
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paper_title: Expander flows, geometric embeddings and graph partitioning

paper_content:

We give a O(slog n)-approximation algorithm for the sparsest cut, edge expansion, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in Rd, whose proof makes essential use of a phenomenon called measure concentration. We also describe an interesting and natural “approximate certificate” for a graph's expansion, which involves embedding an n-node expander in it with appropriate dilation and congestion. We call this an expander flow.
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paper_title: ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT

paper_content:

We show that the Multicut, Sparsest-Cut, and Min-2CNF ? Deletion problems are NP-hard to approximate within every constant factor, assuming the Unique Games Conjecture of Khot (2002). A quantitatively stronger version of the conjecture implies an inapproximability factor of $$\Omega(\sqrt{\log \log n}).$$
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paper_title: DIFFERENTIATING MAPS INTO L 1 , AND THE GEOMETRY OF BV FUNCTIONS

paper_content:

This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps, X-->V, and bi-Lipschitz nonembeddability, where X is a metric measure space and V is a Banach space. Here, we consider the case V=L^1 where differentiability fails. ::: We establish another kind of differentiability for certain X, including R^n and H, the Heisenberg group with its Carnot-Cartheodory metric. It follows that H does not bi-Lipschitz embed into L^1, as conjectured by J. Lee and A. Naor. When combined with their work, this provides a natural counter example to the Goemans-Linial conjecture in theoretical computer science; the first such counterexample was found by Khot-Vishnoi. A key ingredient in the proof of our main theorem is a new connection between Lipschitz maps to L^1 and functions of bounded variation, which permits us to exploit recent work on the structure of BV functions on the Heisenberg group.
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paper_title: Finite metric spaces: combinatorics, geometry and algorithms

paper_content:

In the last several years a number of very interesting results were proved about finite metric spaces. Some of this work is motivated by practical considerations: Large data sets (coming e.g. from computational molecular biology, brain research or data mining) can be viewed as large metric spaces that should be analyzed (e.g. correctly clustered).On the other hand, these investigations connect to some classical areas of geometry - the asymptotic theory of finite-dimensional normed spaces and differential geometry. Finally, the metric theory of finite graphs has proved very useful in the study of graphs per se and the design of approximation algorithms for hard computational problems. In this talk I will try to explain some of the results and review some of the emerging new connections and the many fascinating open problems in this area.
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paper_title: Semidefinite Programming in Combinatorial Optimization

paper_content:

We discuss the use of semidefinite programming for combinatorial optimization problems. The main topics covered include (i) the Lovasz theta function and its applications to stable sets, perfect graphs, and coding theory, (ii) the automatic generation of strong valid inequalities, (iii) the maximum cut problem and related problems, and (iv) the embedding of finite metric spaces and its relationship to the sparsest cut problem.
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paper_title: Integrality Gaps for Strong SDP Relaxations of UNIQUE GAMES

paper_content:

With the work of Khot and Vishnoi (FOCS 2005) as a starting point, we obtain integrality gaps for certain strong SDP relaxations of unique games. Specifically, we exhibit a gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner products of up to $\exp(\Omega(\log\log~n)^{1/4})$ vectors. For stronger relaxations obtained from the basic semidefinite program by $R$ rounds of Sherali--Adams lift-and-project, we prove a unique games integrality gap for $R = \Omega(\log\log~n)^{1/4}$.By composing these SDP gaps with UGC-hardness reductions, the above results imply corresponding integrality gaps for every problem for which a UGC-based hardness is known. Consequently, this work implies that including any valid constraints on up to$\exp(\Omega(\log\log~n)^{1/4})$ vectors to natural semidefinite program, does not improve the approximation ratio for any problem in the following classes: constraint satisfaction problems, ordering constraint satisfaction problems and metric labeling problems over constant-size metrics. We obtain similar SDP integrality gaps for balanced separator, building on Devanur et al. (STOC 2006). We also exhibit, for explicit constants $\gamma, \delta ≫ 0$, an n-point negative-type metric which requires distortion $\Omega(\log\log n)^{\gamma}$ to embed into$\ell_1$, although all its subsets of size$\exp(\Omega(\log\log~n)^{\delta})$ embed isometrically into $\ell_1$.
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paper_title: Lp metrics on the Heisenberg group and the Goemans-Linial conjecture

paper_content:

We prove that the function d : \mathbb{R}^3 \times \mathbb{R}^3 \to [0,\infty ) given by d\left( {(x,y,z),(t,u,v)} \right)= \left( {[((t - x)^2 + (u - y)^2 )^2 + (v - z + 2xu - 2yt)^2 ]^{\frac{1} {2}} + (t - x)^2 + (u - y)^2 } \right)^{\frac{1} {2}} . is a metric on \mathbb{R}^3 such that (\mathbb{R}^3, \sqrt d ) is isometric to a subset of Hilbert space, yet (\mathbb{R}^3, d) does not admit a bi-Lipschitz embedding into L_1. This yields a new simple counter example to the Goemans-Linial conjecture on the integrality gap of the semidefinite relaxation of the Sparsest Cut problem. The metric above is doubling, and hence has a padded stochastic decomposition at every scale. We also study the L_p version of this problem, and obtain a counter example to a natural generalization of a classical theorem of Bretagnolle, Dacunha-Castelle and Krivine (of which the Goemans-Linial conjecture is a particular case). Our methods involve Fourier analytic techniques, and a recent breakthrough of Cheeger and Kleiner, together with classical results of Pansu on the differentiability of Lipschitz functions on the Heisenberg group.
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paper_title: The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into ℓ1

paper_content:

In this paper, we disprove a conjecture of Goemans and Linial; namely, that every negative type metric embeds into $\ell_1$ with constant distortion. We show that for an arbitrarily small constant $\delta> 0$, for all large enough $n$, there is an $n$-point negative type metric which requires distortion at least $(\log\log n)^{1/6-\delta}$ to embed into $\ell_1.$ ::: Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot, establishing a previously unsuspected connection between probabilistically checkable proof systems (PCPs) and the theory of metric embeddings. We first prove that the UGC implies a super-constant hardness result for the (non-uniform) Sparsest Cut problem. Though this hardness result relies on the UGC, we demonstrate, nevertheless, that the corresponding PCP reduction can be used to construct an "integrality gap instance" for Sparsest Cut. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of Unique Games. Then we "simulate" the PCP reduction and "translate" the integrality gap instance of Unique Games to an integrality gap instance of Sparsest Cut. This enables us to prove a $(\log \log n)^{1/6-\delta}$ integrality gap for Sparsest Cut, which is known to be equivalent to the metric embedding lower bound.
---
paper_title: On the differentiation of Lipschitz maps from metric measure spaces to Banach spaces

paper_content:

A sodium chlorite bath for the bleaching of textile fibers, e.g. cotton, is activated by one or more bisulfite derivatives of an organic-reducing compound having one or more aldehyde or ketone functional groups and which are capable of forming bisulfitic combination addition compounds with an alkali bisulfite. Such activators have been found to be effective in bringing about the progressive decomposition of the sodium chlorite of the bath and hence controlled bleaching at relatively low temperatures of the sodium chlorite bath.
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paper_title: L_1 embeddings of the Heisenberg group and fast estimation of graph isoperimetry

paper_content:

We survey connections between the theory of bi-Lipschitz embeddings and the Sparsest Cut Problem in combinatorial optimization. The story of the Sparsest Cut Problem is a striking example of the deep interplay between analysis, geometry, and probability on the one hand, and computational issues in discrete mathematics on the other. We explain how the key ideas evolved over the past 20 years, emphasizing the interactions with Banach space theory, geometric measure theory, and geometric group theory. As an important illustrative example, we shall examine recently established connections to the the structure of the Heisenberg group, and the incompatibility of its Carnot-Carath\'eodory geometry with the geometry of the Lebesgue space $L_1$.
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paper_title: Compression bounds for Lipschitz maps from the Heisenberg group to $L_1$

paper_content:

We prove a quantitative bi-Lipschitz nonembedding theorem for the Heisenberg group with its Carnot-Carath\'eodory metric and apply it to give a lower bound on the integrality gap of the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem.
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paper_title: Spherical Cubes and Rounding in High Dimensions

paper_content:

What is the least surface area of a shape that tiles Ropfd under translations by Zopfd? Any such shape must have volume 1 and hence surface area at least that of the volume-1 ball, namely Omega(radicd). Our main result is a construction with surface area O(radicd), matching the lower bound up to a constant factor of 2radic2pi/eap3. The best previous tile known was only slightly better than the cube, having surface area on the order of d. We generalize this to give a construction that tiles Ropfd by translations of any full rank discrete lattice Lambda with surface area 2piparV-1parfb, where V is the matrix of basis vectors of Lambda, and par.parfb denotes the Frobenius norm. We show that our bounds are optimal within constant factors for rectangular lattices. Our proof is via a random tessellation process, following recent ideas of Raz in the discrete setting. Our construction gives an almost optimal noise-resistant rounding scheme to round points in Ropfd to rectangular lattice points.
---
paper_title: Understanding Parallel Repetition Requires Understanding Foams

paper_content:

Motivated by the study of parallel repetition and also by the unique games conjecture, we investigate the value of the "odd cycle games" under parallel repetition. Using tools from discrete harmonic analysis, we show that after d rounds on the cycle of length m, the value of the game is at most 1-(1/m)ldrOmega macr(radicd) (for dlesm2, say). This beats the natural barrier of 1-Theta(1/m)2 ldrd for Raz-style proofs and also the SDP bound of Feige-Lovasz; however, it just barely fails to have implications for unique games. On the other hand, we also show that improving our bound would require proving nontrivial lower bounds on the surface area of high-dimensional foams. Specifically, one would need to answer: what is the least surface area of a cell that tiles Rd by the lattice Zd?
---
paper_title: Economical toric spines via Cheeger's Inequality

paper_content:

Let $G_{\infty}=(C_m^d)_{\infty}$ denote the graph whose set of vertices is $\{1,..., m\}^d$, where two distinct vertices are adjacent iff they are either equal or adjacent in $C_m$ in each coordinate. Let $G_{1}=(C_m^d)_1$ denote the graph on the same set of vertices in which two vertices are adjacent iff they are adjacent in one coordinate in $C_m$ and equal in all others. Both graphs can be viewed as graphs of the $d$-dimensional torus. We prove that one can delete $O(\sqrt d m^{d-1})$ vertices of $G_1$ so that no topologically nontrivial cycles remain. This improves an $O(d^{\log_2 (3/2)}m^{d-1})$ estimate of Bollob\'as, Kindler, Leader and O'Donnell. We also give a short proof of a result implicit in a recent paper of Raz: one can delete an $O(\sqrt d/m)$ fraction of the edges of $G_{\infty}$ so that no topologically nontrivial cycles remain in this graph. Our technique also yields a short proof of a recent result of Kindler, O'Donnell, Rao and Wigderson; there is a subset of the continuous $d$-dimensional torus of surface area $O(\sqrt d)$ that intersects all nontrivial cycles. All proofs are based on the same general idea: the consideration of random shifts of a body with small boundary and no- nontrivial cycles, whose existence is proved by applying the isoperimetric inequality of Cheeger or its vertex or edge discrete analogues.
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paper_title: On Systems of Linear Equations with Two Variables per Equation

paper_content:

For a prime p, max-2linp is the problem of satisfying as many equations as possible from a system of linear equations modulo p, where every equation contains two variables. Hastad shows that this problem is NP-hard to approximate within a ratio of 11/12 + e for p=2, and Andersson, Engebretsen and Hastad show the same hardness of approximation ratio for p ≥ 11, and somewhat weaker results (such as 69/70) for p = 3,5,7. We prove that max-2linp is easiest to approximate when p = 2, implying for every prime p that max-2linp is NP-hard to approximate within a ratio of 11/12 + e. For large p, we prove stronger hardness of approximation results. Namely, we show that there is some universal constant δ > 0 such that it is NP-hard to approximate max-2linp within a ratio better than 1/p δ . We use our results so as to clarify some aspects of Khot’s unique games conjecture. Namely, we show that for every e > 0 it is NP-hard to approximate the value of unique games within a ratio of e.
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paper_title: Approximate Lasserre Integrality Gap for Unique Games

paper_content:

In this paper, we investigate whether a constant round Lasserre Semi-definite Programming (SDP) relaxation might give a good approximation to the Unique Games problem. We show that the answer is negative if the relaxation is insensitive to a sufficiently small perturbation of the constraints. Specifically, we construct an instance of Unique Games with k labels along with an approximate vector solution to t rounds of the Lasserre SDP relaxation. The SDP objective is at least 1 - e whereas the integral optimum is at most γ, and all SDP constraints are satisfied up to an accuracy of δ > 0. Here e, γ > 0 and t ∈ Z+ are arbitrary constants and k = k(e, γ) ∈ Z+. The accuracy parameter δ can bemade sufficiently small independent of parameters e, γ,t, k (but the size of the instance grows as δ gets smaller).
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Title: On the Unique Games Conjecture (Invited Survey)

Section 1: INTRODUCTION
Description 1: This section introduces the Unique Games Conjecture (UGC), exploring its significance in computational complexity, algorithms, and other areas. It also mentions connections to inapproximability, discrete Fourier analysis, geometry, integrality gaps, algorithms, and parallel repetition.

Section 2: Approximation Algorithms and Inapproximability
Description 2: This section discusses the role of approximation algorithms in solving NP-complete problems, defining approximation factors and the impact of gap-preserving reductions. It includes subsections on the PCP Theorem and optimal inapproximability results.

Section 3: The Unique Games Conjecture
Description 3: This section elaborates on the Unique Games Conjecture itself, its definition, and some key observations. It also mentions the conjecture's implications and details flow to formalize certain properties.

Section 4: Boolean Functions, Dictatorships, and Influence of Variables
Description 4: Here, the concepts of boolean functions, dictatorship, and variable influence are described. It discusses how these play critical roles in the context of UGC-based inapproximability results.

Section 5: Integrality Gaps
Description 5: This section delves into how integrality gaps hinder polynomial time algorithms' ability to provide good approximate solutions. It includes a discussion on the MaxCut problem and the strategies for constructing explicit integrality gap instances.

Section 6: VARIANTS OF THE Unique Games Conjecture
Description 6: Different variants of the Unique Games Conjecture that are useful for proving inapproximability results are described in this section. It also discusses the equivalence of certain conjectures to the UGC.

Section 7: INAPPROXIMABILITY RESULTS
Description 7: This section lists main inapproximability results motivated by UGC, characterizing the exact approximation thresholds. It includes a sketch of a reduction from Unique Game problem to Min-2SAT-Deletion and discusses Raghavendra's result.

Section 8: ALGORITHMS
Description 8: This section summarizes algorithmic results related to UGC, detailing methods for approximating solutions to Unique Games and discussing the limitations of these algorithms.

Section 9: DISCRETE FOURIER ANALYSIS
Description 9: The importance of Fourier analysis in deducing inapproximability results is discussed in this section. Various Fourier analytic theorems and their applications to inapproximability are covered.

Section 10: VARIANTS OF THE 2-Prover-1-Round Game
Description 10: This section discusses generalizations and variants of the 2-Prover-1-Round games, including the Unique Game problem and related parallel repetition theorems.

Section 11: GEOMETRY
Description 11: This section explores the geometric aspects associated with the Unique Games Conjecture, discussing connections to Fourier analytic theorems and integrality gaps. It also touches on constructing foams using tiling of space.

Section 12: CONCLUSION
Description 12: The paper concludes with arguments for and against the Unique Games Conjecture, summarizing the current state of knowledge, its implications, and potential future research directions.