A Survey of Heuristics for Domain-Independent Planning
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paper_title: Complexity results for standard benchmark domains in planning

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The efficiency of AI planning systems is usually evaluated empirically. For the validity of conclusions drawn from such empirical data, the problem set used for evaluation is of critical importance. In planning, this problem set usually, or at least often, consists of tasks from the various planning domains used in the first two international planning competitions, hosted at the 1998 and 2000 AIPS conferences. It is thus surprising that comparatively little is known about the properties of these benchmark domains, with the exception of BLOCKSWORLD, which has been studied extensively by several research groups.In this contribution, we try to remedy this fact by providing a map of the computational complexity of non-optimal and optimal planning for the set of domains used in the competitions. We identify a common transportation theme shared by the majority of the benchmarks and use this observation to define and analyze a general transportation problem that generalizes planning in several classical domains such as LOGISTICS, MYSTERY and GRIPPER. We then apply the results of that analysis to the actual transportation domains from the competitions. We next examine the remaining benchmarks, which do not exhibit a strong transportation feature, namely SCHEDULE and FREECELL.Relating the results of our analysis to empirical work on the behavior of the recently very successful FF planning system, we observe that our theoretical results coincide well with data obtained from empirical investigations.
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paper_title: LPG: A planner based on local search for planning graphs

paper_content:

We present LPG, a fast planner using local search for solving planning graphs. LPG can use various heuristics based on a parametrized objective function. These parameters weight different types of inconsistencies in the partial plan represented by the current search state, and are dynamically evaluated during search using Lagrange multipliers. LPG's basic heuristic was inspired by Walksat, which in Kautz and Selman's Blackbox can be used to solve the SAT-encoding of a planning graph. An advantage of LPG is that its heuristics exploit the structure of the planning graph, while Black-box relies on general heuristics for SAT-problems, and requires the translation of the planning graph into propositional clauses. Another major difference is that LPG can handle action costs to produce good quality plans. This is achieved by an "anytime" process minimizing an objective function based on the number of inconsistencies in the partial plan and on its overall cost. The objective function can also take into account the number of parallel steps and the overall plan duration. Experimental results illustrate the efficiency of our approach showing, in particular, that for a set of well-known benchmark domains LPG is significantly faster than existing Graphplan-style planners.
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paper_title: The Fast Downward Planning System

paper_content:

Fast Downward is a classical planning system based on heuristic search. It can deal with general deterministic planning problems encoded in the propositional fragment of PDDL2.2, including advanced features like ADL conditions and effects and derived predicates (axioms). Like other well-known planners such as HSP and FF, Fast Downward is a progression planner, searching the space of world states of a planning task in the forward direction. However, unlike other PDDL planning systems, Fast Downward does not use the propositional PDDL representation of a planning task directly. Instead, the input is first translated into an alternative representation called multi-valued planning tasks, which makes many of the implicit constraints of a propositional planning task explicit. Exploiting this alternative representation, Fast Downward uses hierarchical decompositions of planning tasks for computing its heuristic function, called the causal graph heuristic, which is very different from traditional HSP-like heuristics based on ignoring negative interactions of operators. In this article, we give a full account of Fast Downwards approach to solving multi-valued planning tasks. We extend our earlier discussion of the causal graph heuristic to tasks involving axioms and conditional effects and present some novel techniques for search control that are used within Fast Downwards best-first search algorithm: preferred operators transfer the idea of helpful actions from local search to global best-first search, deferred evaluation of heuristic functions mitigates the negative effect of large branching factors on search performance, and multi-heuristic best-first search combines several heuristic evaluation functions within a single search algorithm in an orthogonal way. We also describe efficient data structures for fast state expansion (successor generators and axiom evaluators) and present a new non-heuristic search algorithm called focused iterative-broadening search, which utilizes the information encoded in causal graphs in a novel way. Fast Downward has proven remarkably successful: It won the"classical (i.e., propositional, non-optimising) track of the 4th International Planning Competition at ICAPS 2004, following in the footsteps of planners such as FF and LPG. Our experiments show that it also performs very well on the benchmarks of the earlier planning competitions and provide some insights about the usefulness of the new search enhancements.
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paper_title: Planning as Heuristic Search

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In the AIPS98 Planning Contest, the HSP planner showed that heuristic search planners can be competitive with state-of-the-art Graphplan and SAT planners. Heuristic search planners like HSP transform planning problems into problems of heuristic search by automatically extracting heuristics from Strips encodings. They differ from specialized problem solvers such as those developed for the 24-Puzzle and Rubik’s Cube in that they use a general declarative language for stating problems and a general mechanism for extracting heuristics from these representations. In this paper, we study a family of heuristic search planners that are based on a simple and general heuristic that assumes that action preconditions are independent. The heuristic is then used in the context of best-first and hill-climbing search algorithms, and is tested over a large collection of domains. We then consider variations and extensions such as reversing the direction of the search for speeding node evaluation, and extracting information about propositional invariants for avoiding dead-ends. We analyze the resulting planners, evaluate their performance, and explain when they do best. We also compare the performance of these planners with two state-of-the-art planners, and show that the simplest planner based on a pure best-first search yields the most solid performance over a large set of problems. We also discuss the strengths and limitations of this approach, establish a correspondence between heuristic search planning and Graphplan, and briefly survey recent ideas that can reduce the current gap in performance between general heuristic search planners and specialized solvers.  2001 Elsevier Science B.V. All rights reserved.
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paper_title: STRIPS: A New Approach to the Application of Theorem Proving to Problem Solving

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We describe a new problem solver called STRIPS that attempts to find a sequence of operators in a space of world models to transform a given initial world model into a model in which a given goal formula can be proven to be true. STRIPS represents a world model as an arbi trary collection of first-order predicate calculus formulas and is designed to work with models consisting of large numbers of formulas. It employs a resolution theorem prover to answer (juestions of particular models and uses means-ends analysis to guide it to the desired goal-satisfying model.
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paper_title: An approach to efficient planning with numerical fluents and multi-criteria plan quality

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Dealing with numerical information is practically important in many real-world planning domains where the executability of an action can depend on certain numerical conditions, and the action effects can consume or renew some critical continuous resources, which in pddl can be represented by numerical fluents. When a planning problem involves numerical fluents, the quality of the solutions can be expressed by an objective function that can take different plan quality criteria into account. We propose an incremental approach to automated planning with numerical fluents and multi-criteria objective functions for pddl numerical planning problems. The techniques in this paper significantly extend the framework of planning with action graphs and local search implemented in the lpg planner. We define the numerical action graph (NA-graph) representation for numerical plans and we propose some new local search techniques using this representation, including a heuristic search neighborhood for NA-graphs, a heuristic evaluation function based on relaxed numerical plans, and an incremental method for plan quality optimization based on particular search restarts. Moreover, we analyze our approach through an extensive experimental study aimed at evaluating the importance of some specific techniques for the performance of the approach, and at analyzing its effectiveness in terms of fast computation of a valid plan and quality of the best plan that can be generated within a given CPU-time limit. Overall, the results show that our planner performs quite well compared to other state-of-the-art planners handling numerical fluents.
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paper_title: Cost-Optimal Planning with Landmarks

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Planning landmarks are facts that must be true at some point in every solution plan. Previous work has very successfully exploited planning landmarks in satisficing (non-optimal) planning. We propose a methodology for deriving admissible heuristic estimates for cost-optimal planning from a set of planning landmarks. The resulting heuristics fall into a novel class of multi-path dependent heuristics, and we present a simple best-first search procedure exploiting such heuristics. Our empirical evaluation shows that this framework favorably competes with the state-of-the-art of cost-optimal heuristic search.
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paper_title: The FF Planning System: Fast Plan Generation Through Heuristic Search

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We describe and evaluate the algorithmic techniques that are used in the FF planning system. Like the HSP system, FF relies on forward state space search, using a heuristic that estimates goal distances by ignoring delete lists. Unlike HSP's heuristic, our method does not assume facts to be independent. We introduce a novel search strategy that combines hill-climbing with systematic search, and we show how other powerful heuristic information can be extracted and used to prune the search space. FF was the most successful automatic planner at the recent AIPS-2000 planning competition. We review the results of the competition, give data for other benchmark domains, and investigate the reasons for the runtime performance of FF compared to HSP.
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paper_title: The LAMA Planner: Guiding Cost-Based Anytime Planning with Landmarks

paper_content:

LAMA is a classical planning system based on heuristic forward search. Its core feature is the use of a pseudo-heuristic derived from landmarks, propositional formulas that must be true in every solution of a planning task. LAMA builds on the Fast Downward planning system, using finite-domain rather than binary state variables and multi-heuristic search. The latter is employed to combine the landmark heuristic with a variant of the well-known FF heuristic. Both heuristics are cost-sensitive, focusing on high-quality solutions in the case where actions have non-uniform cost. A weighted A* search is used with iteratively decreasing weights, so that the planner continues to search for plans of better quality until the search is terminated. ::: ::: LAMA showed best performance among all planners in the sequential satisficing track of the International Planning Competition 2008. In this paper we present the system in detail and investigate which features of LAMA are crucial for its performance. We present individual results for some of the domains used at the competition, demonstrating good and bad cases for the techniques implemented in LAMA. Overall, we find that using landmarks improves performance, whereas the incorporation of action costs into the heuristic estimators proves not to be beneficial. We show that in some domains a search that ignores cost solves far more problems, raising the question of how to deal with action costs more effectively in the future. The iterated weighted A* search greatly improves results, and shows synergy effects with the use of landmarks.
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paper_title: Planning with Pattern Databases

paper_content:

Heuristic search planning effectively finds solutions for large planning problems, but since the estimates are either not admissible or too weak, optimal solutions are found in rare cases only. In contrast, heuristic pattern databases are known to significantly improve lower bound estimates for optimally solving challenging single-agent problems like the 24-Puzzle or Rubik’s Cube. This paper studies the effect of pattern databases in the context of deterministic planning. Given a fixed state description based on instantiated predicates, we provide a general abstraction scheme to automatically create admissible domain-independent memory-based heuristics for planning problems, where abstractions are found in factorizing the planning space. We evaluate the impact of pattern database heuristics in A* and hill climbing algorithms for a collection of benchmark domains.
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paper_title: Speeding up the calculation of heuristics for heuristic search-based planning

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Heuristic search-based planners, such as HSP 2.0, solve STRIPS-style planning problems efficiently but spend about eighty percent of their planning time on calculating the heuristic values. In this paper, we systematically evaluate alternative methods for calculating the heuristic values for HSP 2.0 and demonstrate that the resulting planning times differ substantially. HSP 2.0 calculates each heuristic value by solving a relaxed planning problem with a dynamic programming method similar to value iteration. We identify two different approaches for speeding up the calculation of heuristic values, namely to order the value updates and to reuse information from the calculation of previous heuristic values. We then show how these two approaches can be combined, resulting in our PINCH method. PINCH outperforms both of the other approaches individually as well as the methods used by HSP 1.0 and HSP 2.0 for most of the large planning problems tested. In fact, it speeds up the planning time of HSP 2.0 by up to eighty percent in several domains and, in general, the amount of savings grows with the size of the domains, allowing HSP 2.0 to solve larger planning problems than was possible before in the same amount of time and without changing its overall operation.
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paper_title: Fast Planning Through Planning Graph Analysis

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Abstract We introduce a new approach to planning in STRIPS-like domains based on constructing and analyzing a compact structure we call a planning graph. We describe a new planner, Graphplan, that uses this paradigm. Graphplan always returns a shortest possible partial-order plan, or states that no valid plan exists. We provide empirical evidence in favor of this approach, showing that Graphplan outperforms the total-order planner, Prodigy and the partial-order planner, UCPOP, on a variety of interesting natural and artificial planning problems. We also give empirical evidence that the plans produced by Graphplan are quite sensible. Since searches made by this approach are fundamentally different from the searches of other common planning methods, they provide a new perspective on the planning problem.
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paper_title: The FF Planning System: Fast Plan Generation Through Heuristic Search

paper_content:

We describe and evaluate the algorithmic techniques that are used in the FF planning system. Like the HSP system, FF relies on forward state space search, using a heuristic that estimates goal distances by ignoring delete lists. Unlike HSP's heuristic, our method does not assume facts to be independent. We introduce a novel search strategy that combines hill-climbing with systematic search, and we show how other powerful heuristic information can be extracted and used to prune the search space. FF was the most successful automatic planner at the recent AIPS-2000 planning competition. We review the results of the competition, give data for other benchmark domains, and investigate the reasons for the runtime performance of FF compared to HSP.
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paper_title: Learning heuristic functions from relaxed plans

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We present a novel approach to learning heuristic functions for AI planning domains. Given a state, we view a relaxed plan (RP) found from that state as a relational database, which includes the current state and goal facts, the actions in the RP, and the actions' add and delete lists. We represent heuristic functions as linear combinations of generic features of the database, selecting features and weights using training data from solved problems in the target planning domain. Many recent competitive planners use RP-based heuristics, but focus exclusively on the length of the RP, ignoring other RP features. Since RP construction ignores delete lists, for many domains, RP length dramatically under-estimates the distance to a goal, providing poor guidance. By using features that depend on deleted facts and other RP properties, our learned heuristics can potentially capture patterns that describe where such under-estimation occurs. Experiments in the STRIPS domains of IPC 3 and 4 show that best-first search using the learned heuristic can outperform FF (Hoffmann & Nebel 2001), which provided our training data, and frequently outperforms the top performances in IPC 4.
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paper_title: Admissible Heuristics for Optimal Planning

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HSP and HSPr are two recent planners that search the state-space using an heuristic function extracted from Strips encodings. HSP does a forward search from the initial state recomputing the heuristic in every state, while HSPr does a regression search from the goal computing a suitable representation of the heuristic only once. Both planners have shown good performance, often producing solutions that are competitive in time and number of actions with the solutions found by Graphplan and SAT planners. HSP and HSPr. however, are not optimal planners. This is because the heuristic function is not admissible and the search algorithms are not optimal. In this paper we address this problem. We formulate a new admissible heuristic for planning, use it to guide an IDA* search, and empirically evaluate the resulting optimal planner over a number of domains. ::: ::: The main contribution is the idea underlying the heuristic that yields not one but a whole family of polynomial and admissible heuristics that trade accuracy for efficiency. The formulation is general and sheds some light on the heuristics used in HSP and Graphplan, and their relation. It exploits the factored (Strips) representation of planning problems, mapping shortest-path problems in state-space into suitably defined shortest-path problems in atom-space. The formulation applies with little variation to sequential and parallel planning, and problems with different action costs.
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paper_title: Landmarks, critical paths and abstractions: What’s the difference anyway

paper_content:

Current heuristic estimators for classical domain-independent planning are usually based on one of four ideas: delete relaxations, critical paths, abstractions, and, most recently, landmarks. Previously, these different ideas for deriving heuristic functions were largely unconnected. ::: ::: We prove that admissible heuristics based on these ideas are in fact very closely related. Exploiting this relationship, we introduce a new admissible heuristic called the landmark cut heuristic, which compares favourably with the state of the art in terms of heuristic accuracy and overall performance.
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paper_title: New Admissible Heuristics for Domain-Independent Planning

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Admissible heuristics are critical for effective domain-independent planning when optimal solutions must be guaranteed. Two useful heuristics are the hm heuristics, which generalize the reachability heuristic underlying the planning graph, and pattern database heuristics. These heuristics, however, have serious limitations: reachability heuristics capture only the cost of critical paths in a relaxed problem, ignoring the cost of other relevant paths, while PDB heuristics, additive or not, cannot accommodate too many variables in patterns, and methods for automatically selecting patterns that produce good estimates are not known. ::: ::: We introduce two refinements of these heuristics: First, the additive hm heuristic which yields an admissible sum of hm heuristics using a partitioning of the set of actions. Second, the constrained PDB heuristic which uses constraints from the original problem to strengthen the lower bounds obtained from abstractions. ::: ::: The new heuristics depend on the way the actions or problem variables are partitioned. We advance methods for automatically deriving additive hm and PDB heuristics from STRIPS encodings. Evaluation shows improvement over existing heuristics in several domains, although, not surprisingly, no heuristic dominates all the others over all domains.
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paper_title: The Fast Downward Planning System

paper_content:

Fast Downward is a classical planning system based on heuristic search. It can deal with general deterministic planning problems encoded in the propositional fragment of PDDL2.2, including advanced features like ADL conditions and effects and derived predicates (axioms). Like other well-known planners such as HSP and FF, Fast Downward is a progression planner, searching the space of world states of a planning task in the forward direction. However, unlike other PDDL planning systems, Fast Downward does not use the propositional PDDL representation of a planning task directly. Instead, the input is first translated into an alternative representation called multi-valued planning tasks, which makes many of the implicit constraints of a propositional planning task explicit. Exploiting this alternative representation, Fast Downward uses hierarchical decompositions of planning tasks for computing its heuristic function, called the causal graph heuristic, which is very different from traditional HSP-like heuristics based on ignoring negative interactions of operators. In this article, we give a full account of Fast Downwards approach to solving multi-valued planning tasks. We extend our earlier discussion of the causal graph heuristic to tasks involving axioms and conditional effects and present some novel techniques for search control that are used within Fast Downwards best-first search algorithm: preferred operators transfer the idea of helpful actions from local search to global best-first search, deferred evaluation of heuristic functions mitigates the negative effect of large branching factors on search performance, and multi-heuristic best-first search combines several heuristic evaluation functions within a single search algorithm in an orthogonal way. We also describe efficient data structures for fast state expansion (successor generators and axiom evaluators) and present a new non-heuristic search algorithm called focused iterative-broadening search, which utilizes the information encoded in causal graphs in a novel way. Fast Downward has proven remarkably successful: It won the"classical (i.e., propositional, non-optimising) track of the 4th International Planning Competition at ICAPS 2004, following in the footsteps of planners such as FF and LPG. Our experiments show that it also performs very well on the benchmarks of the earlier planning competitions and provide some insights about the usefulness of the new search enhancements.
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paper_title: Optimal admissible composition of abstraction heuristics

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Additive ensembles of admissible heuristics constitute the most general form of exploiting the individual strengths of numerous admissible heuristics in optimal planning. However, the same set of heuristics can be additively composed in infinitely many ways and the quality of the resulting heuristic estimate depends directly on the choice of the composition. Focusing on abstraction heuristics, we describe a procedure that takes a deterministic planning problem, a forward-search state, and a set of abstraction-based admissible heuristics, and derives an optimal additive composition of these heuristics with respect to the given state. Most importantly, we show that this procedure is polynomial-time for arbitrary sets of all abstraction heuristics with which we are acquainted, including explicit abstractions such as pattern databases (regular or constrained) and merge-and-shrink, and implicit abstractions such as fork-decomposition and abstractions based on tractable constraint optimization over tree-shaped constraint networks.
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paper_title: Disjoint pattern database heuristics

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We describe a new technique for designing more accurate admissible heuristic evaluation functions, based on pattern databases [J. Culberson, J. Schaeffer, Comput. Intelligence 14 (3) (1998) 318-334]. While many heuristics, such as Manhattan distance, compute the cost of solving individual subgoals independently, pattern databases consider the cost of solving multiple subgoals simultaneously. Existing work on pattern databases allows combining values from different pattern databases by taking their maximum. If the subgoals can be divided into disjoint subsets so that each operator only affects subgoals in one subset, then we can add the pattern-database values for each subset, resulting in a more accurate admissible heuristic function. We used this technique to improve performance on the Fifteen Puzzle by a factor of over 2000, and to find optimal solutions to 50 random instances of the Twenty-Four Puzzle.
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paper_title: Explicitstate abstraction: A new method for generating heuristic functions

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Many AI problems can be recast as finding an optimal path in a discrete state space. An abstraction defines an admissible heuristic function as the distances in a smaller state space where arbitrary sets of states are "aggregated" into single states. A special case are pattern database (PDB) heuristics, which aggregate states if they agree on the state variables inside the pattern. Explicit-state abstraction is more flexible, explicitly aggregating selected pairs of states in a process that interleaves composition of abstractions with abstraction of the composites. The increased flexibility gains expressive power: sometimes, the real cost function can be represented concisely as an explicit-state abstraction, but not as a PDB. Explicit-state abstraction has been applied to planning and model checking, with highly promiSing empirical results.
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paper_title: Implicit Abstraction Heuristics

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State-space search with explicit abstraction heuristics is at the state of the art of cost-optimal planning. These heuristics are inherently limited, nonetheless, because the size of the abstract space must be bounded by some, even if a very large, constant. Targeting this shortcoming, we introduce the notion of (additive) implicit abstractions, in which the planning task is abstracted by instances of tractable fragments of optimal planning. We then introduce a concrete setting of this framework, called fork-decomposition, that is based on two novel fragments of tractable cost-optimal planning. The induced admissible heuristics are then studied formally and empirically. This study testifies for the accuracy of the fork decomposition heuristics, yet our empirical evaluation also stresses the tradeoff between their accuracy and the runtime complexity of computing them. Indeed, some of the power of the explicit abstraction heuristics comes from precomputing the heuristic function offine and then determining h(s) for each evaluated state s by a very fast lookup in a "database." By contrast, while fork-decomposition heuristics can be calculated in polynomial time, computing them is far from being fast. To address this problem, we show that the time-per-node complexity bottleneck of the fork-decomposition heuristics can be successfully overcome. We demonstrate that an equivalent of the explicit abstraction notion of a "database" exists for the fork-decomposition abstractions as well, despite their exponential-size abstract spaces. We then verify empirically that heuristic search with the "databased" fork-decomposition heuristics favorably competes with the state of the art of cost-optimal planning.
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paper_title: Flexible abstraction heuristics for optimal sequential planning

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We describe an approach to deriving consistent heuristics for automated planning, based on explicit search in abstract state spaces. The key to managing complexity is interleaving composition of abstractions over different sets of state variables with abstraction of the partial composites. ::: ::: The approach is very general and can be instantiated in many different ways by following different abstraction strategies. In particular, the technique subsumes planning with pattern databases as a special case. Moreover, with suitable abstraction strategies it is possible to derive perfect heuristics in a number of classical benchmark domains, thus allowing their optimal solution in polynomial time. ::: ::: To evaluate the practical usefulness of the approach, we perform empirical experiments with one particular abstraction strategy. Our results show that the approach is competitive with the state of the art.
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paper_title: Domain-independent construction of pattern database heuristics for cost-optimal planning

paper_content:

Heuristic search is a leading approach to domain-independent planning. For cost-optimal planning, however, existing admissible heuristics are generally too weak to effectively guide the search. Pattern database heuristics (PDBs), which are based on abstractions of the search space, are currently one of the most promising approaches to developing better admissible heuristics. The informedness of PDB heuristics depends crucially on the selection of appropriate abstractions (patterns). Although PDBs have been applied to many search problems, including planning, there are not many insights into how to select good patterns, even manually. What constitutes a good pattern depends on the problem domain, making the task even more difficult for domain-independent planning, where the process needs to be completely automatic and generaL We present a novel way of constructing good patterns automatically from the specification of planning problem instances. We demonstrate that this allows a domain-independent planner to solve planning problems optimally in some very challenging domains, including a STRIPS formulation of the Sokoban puzzle.
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paper_title: Structural-pattern databases

paper_content:

Explicit abstraction heuristics, notably pattern-database and merge-and-shrink heuristics, are employed by some state-of-the-art optimal heuristic-search planners. The major limitation of these abstraction heuristics is that the size of the abstract space has to be bounded by a (large) constant. Targeting this issue, Katz and Domshlak (2008b) introduced structural, and in particular fork-decomposition, abstractions, in which the planning task is abstracted by an instance of a tractable fragment of optimal planning. At first view, however, the lunch was not free. Some of the power of the explicit abstraction heuristics comes from pre-computing the heuristic function offline, and then determine h(s) for each evaluated state s by a very fast lookup in a "database". In contrast, fork-decomposition offer a poly-time, yet far from being fast, computation. ::: ::: In this contribution, we show that the time-per-node complexity bottleneck of the fork-decomposition heuristics can be successfully overcome. Specifically, we show that an equivalent of the explicit abstractions' notion of "database" exists for the fork-decomposition abstractions as well, and this despite of their exponential-size abstract spaces. Experimentally, we show that heuristic search with such "databased" fork-decomposition heuristics favorably competes with the state-of-the-art of optimal planning.
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paper_title: Symbolic pattern databases in heuristic search panning

paper_content:

This paper invents symbolic pattern databases (SPDB) to combine two influencing aspects for recent progress in domain-independent action planning, namely heuristic search and model checking. SPDBs are off-line computed dictionaries, generated in symbolic backward traversals of automatically inferred planning space abstractions. ::: ::: The entries of SPDBs serve as heuristic estimates to accelerate explicit and symbolic, approximate and optimal heuristic search planners. Selected experiments highlight that the symbolic representation yields much larger and more accurate pattern databases than the ones generated with explicit methods.
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paper_title: Planning with Pattern Databases

paper_content:

Heuristic search planning effectively finds solutions for large planning problems, but since the estimates are either not admissible or too weak, optimal solutions are found in rare cases only. In contrast, heuristic pattern databases are known to significantly improve lower bound estimates for optimally solving challenging single-agent problems like the 24-Puzzle or Rubik’s Cube. This paper studies the effect of pattern databases in the context of deterministic planning. Given a fixed state description based on instantiated predicates, we provide a general abstraction scheme to automatically create admissible domain-independent memory-based heuristics for planning problems, where abstractions are found in factorizing the planning space. We evaluate the impact of pattern database heuristics in A* and hill climbing algorithms for a collection of benchmark domains.
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paper_title: Finding Optimal Solutions to the Twenty-Four Puzzle

paper_content:

We have found the first optimal solutions to random instances of the Twenty-Four Puzzle, the 5 × 5 version of the well-known sliding-tile puzzles. Our new contribution to this problem is a more powerful admissible heuristic function. We present a general theory for the automatic discovery of such heuristics, which is based on considering multiple subgoals simultaneously. In addition, we apply a technique for pruning duplicate nodes in depth-first search using a finitestate machine. Finally, we observe that as heuristic search problems are scaled up, more powerful heuristic functions become both necessary and cost-effective.
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paper_title: Ordered Landmarks in Planning

paper_content:

Many known planning tasks have inherent constraints concerning the best order in which to achieve the goals. A number of research efiorts have been made to detect such constraints and to use them for guiding search, in the hope of speeding up the planning process. ::: ::: We go beyond the previous approaches by considering ordering constraints not only over the (top-level) goals, but also over the sub-goals that will necessarily arise during planning. Landmarks are facts that must be true at some point in every valid solution plan. We extend Koehler and Hoffmann's definition of reasonable orders between top level goals to the more general case of landmarks. We show how landmarks can be found, how their reasonable orders can be approximated, and how this information can be used to decompose a given planning task into several smaller sub-tasks. Our methodology is completely domain- and planner-independent. The implementation demonstrates that the approach can yield significant runtime performance improvements when used as a control loop around state-of-the-art sub-optimal planning systems, as exemplified by FF and LPG.
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paper_title: Cost-Optimal Planning with Landmarks

paper_content:

Planning landmarks are facts that must be true at some point in every solution plan. Previous work has very successfully exploited planning landmarks in satisficing (non-optimal) planning. We propose a methodology for deriving admissible heuristic estimates for cost-optimal planning from a set of planning landmarks. The resulting heuristics fall into a novel class of multi-path dependent heuristics, and we present a simple best-first search procedure exploiting such heuristics. Our empirical evaluation shows that this framework favorably competes with the state-of-the-art of cost-optimal heuristic search.
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paper_title: The LAMA Planner: Guiding Cost-Based Anytime Planning with Landmarks

paper_content:

LAMA is a classical planning system based on heuristic forward search. Its core feature is the use of a pseudo-heuristic derived from landmarks, propositional formulas that must be true in every solution of a planning task. LAMA builds on the Fast Downward planning system, using finite-domain rather than binary state variables and multi-heuristic search. The latter is employed to combine the landmark heuristic with a variant of the well-known FF heuristic. Both heuristics are cost-sensitive, focusing on high-quality solutions in the case where actions have non-uniform cost. A weighted A* search is used with iteratively decreasing weights, so that the planner continues to search for plans of better quality until the search is terminated. ::: ::: LAMA showed best performance among all planners in the sequential satisficing track of the International Planning Competition 2008. In this paper we present the system in detail and investigate which features of LAMA are crucial for its performance. We present individual results for some of the domains used at the competition, demonstrating good and bad cases for the techniques implemented in LAMA. Overall, we find that using landmarks improves performance, whereas the incorporation of action costs into the heuristic estimators proves not to be beneficial. We show that in some domains a search that ignores cost solves far more problems, raising the question of how to deal with action costs more effectively in the future. The iterated weighted A* search greatly improves results, and shows synergy effects with the use of landmarks.
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paper_title: When Abstractions Met Landmarks

paper_content:

Abstractions and landmarks are two powerful mechanisms for devising admissible heuristics for classical planning. Here we aim at putting them together by integrating landmark information into abstractions, and propose a concrete realization of this direction suitable for structural-pattern abstractions, as well as for other abstraction heuristics. Our empirical evaluation shows that landmark information can substantially improve the quality of abstraction heuristic estimates.
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paper_title: Landmarks, critical paths and abstractions: What’s the difference anyway

paper_content:

Current heuristic estimators for classical domain-independent planning are usually based on one of four ideas: delete relaxations, critical paths, abstractions, and, most recently, landmarks. Previously, these different ideas for deriving heuristic functions were largely unconnected. ::: ::: We prove that admissible heuristics based on these ideas are in fact very closely related. Exploiting this relationship, we introduce a new admissible heuristic called the landmark cut heuristic, which compares favourably with the state of the art in terms of heuristic accuracy and overall performance.
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paper_title: Accuracy of admissible heuristic functions in selected planning domains

paper_content:

The efficiency of optimal planning algorithms based on heuristic search crucially depends on the accuracy of the heuristic function used to guide the search. Often, we are interested in domain-independent heuristics for planning. In order to assess the limitations of domain-independent heuristic planning, we analyze the (in)accuracy of common domain-independent planning heuristics in the IPC benchmark domains. For a selection of these domains, we analytically investigate the accuracy of the h+ heuristic, the hm family of heuristics, and certain (additive) pattern database heuristics, compared to the perfect heuristic h*. Whereas h+and additive pattern database heuristics usually return cost estimates proportional to the true cost, non-additive hm and nonadditive pattern-database heuristics can yield results underestimating the true cost by arbitrarily large factors.
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paper_title: Anytime Heuristic Search for Partial Satisfaction Planning

paper_content:

We present a heuristic search approach to solve partial satisfaction planning (PSP) problems. In these problems, goals are modeled as soft constraints with utility values, and actions have costs. Goal utility represents the value of each goal to the user and action cost represents the total resource cost (e.g., time, fuel cost) needed to execute each action. The objective is to find the plan that maximizes the trade-off between the total achieved utility and the total incurred cost; we call this problem PSP Net Benefit. Previous approaches to solving this problem heuristically convert PSP Net Benefit into STRIPS planning with action cost by pre-selecting a subset of goals. In contrast, we provide a novel anytime search algorithm that handles soft goals directly. Our new search algorithm has an anytime property that keeps returning better quality solutions until the termination criteria are met. We have implemented this search algorithm, along with relaxed plan heuristics adapted to PSP Net Benefit problems, in a forward state-space planner called Sapa^P^S. An adaptation of Sapa^P^S, called Yochan^P^S, received a ''distinguished performance'' award in the ''simple preferences'' track of the 5th International Planning Competition.
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paper_title: An approach to efficient planning with numerical fluents and multi-criteria plan quality

paper_content:

Dealing with numerical information is practically important in many real-world planning domains where the executability of an action can depend on certain numerical conditions, and the action effects can consume or renew some critical continuous resources, which in pddl can be represented by numerical fluents. When a planning problem involves numerical fluents, the quality of the solutions can be expressed by an objective function that can take different plan quality criteria into account. We propose an incremental approach to automated planning with numerical fluents and multi-criteria objective functions for pddl numerical planning problems. The techniques in this paper significantly extend the framework of planning with action graphs and local search implemented in the lpg planner. We define the numerical action graph (NA-graph) representation for numerical plans and we propose some new local search techniques using this representation, including a heuristic search neighborhood for NA-graphs, a heuristic evaluation function based on relaxed numerical plans, and an incremental method for plan quality optimization based on particular search restarts. Moreover, we analyze our approach through an extensive experimental study aimed at evaluating the importance of some specific techniques for the performance of the approach, and at analyzing its effectiveness in terms of fast computation of a valid plan and quality of the best plan that can be generated within a given CPU-time limit. Overall, the results show that our planner performs quite well compared to other state-of-the-art planners handling numerical fluents.
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paper_title: State-Set Branching: Leveraging BDDs for Heuristic Search

paper_content:

In this article, we present a framework called state-set branching that combines symbolic search based on reduced ordered Binary Decision Diagrams (BDDs) with best-first search, such as A* and greedy best-first search. The framework relies on an extension of these algorithms from expanding a single state in each iteration to expanding a set of states. We prove that it is generally sound and optimal for two A* implementations and show how a new BDD technique called branching partitioning can be used to efficiently expand sets of states. The framework is general. It applies to any heuristic function, evaluation function, and transition cost function defined over a finite domain. Moreover, branching partitioning applies to both disjunctive and conjunctive transition relation partitioning. An extensive experimental evaluation of the two A* implementations proves state-set branching to be a powerful framework. The algorithms outperform the ordinary A* algorithm in almost all domains. In addition, they can improve the complexity of A* exponentially and often dominate both A* and blind BDD-based search by several orders of magnitude. Moreover, they have substantially better performance than BDDA*, the currently most efficient BDD-based implementation of A*.
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paper_title: Anytime Heuristic Search for Partial Satisfaction Planning

paper_content:

We present a heuristic search approach to solve partial satisfaction planning (PSP) problems. In these problems, goals are modeled as soft constraints with utility values, and actions have costs. Goal utility represents the value of each goal to the user and action cost represents the total resource cost (e.g., time, fuel cost) needed to execute each action. The objective is to find the plan that maximizes the trade-off between the total achieved utility and the total incurred cost; we call this problem PSP Net Benefit. Previous approaches to solving this problem heuristically convert PSP Net Benefit into STRIPS planning with action cost by pre-selecting a subset of goals. In contrast, we provide a novel anytime search algorithm that handles soft goals directly. Our new search algorithm has an anytime property that keeps returning better quality solutions until the termination criteria are met. We have implemented this search algorithm, along with relaxed plan heuristics adapted to PSP Net Benefit problems, in a forward state-space planner called Sapa^P^S. An adaptation of Sapa^P^S, called Yochan^P^S, received a ''distinguished performance'' award in the ''simple preferences'' track of the 5th International Planning Competition.
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Title: A Survey of Heuristics for Domain-Independent Planning

Section 1: INTRODUCTION
Description 1: This section introduces the context and motivation for the survey, discusses the evolution of heuristic search-based planners, and outlines the various types of heuristics utilized in state space planning and other paradigms.

Section 2: HEURISTICS FOR STATE-SPACE PLANNING
Description 2: This section discusses the concept of heuristics and briefly reviews the STRIPS planning task, focusing on several classes of heuristics used in state space planning.

Section 3: Heuristic
Description 3: This subsection defines the notion of heuristics, discussing strategies and problem types where heuristics are applicable, and the importance of admissible and inadmissible heuristics.

Section 4: STRIPS Planning Task
Description 4: This subsection explains the STRIPS planning model and details the additive and maximum heuristics, discussing their applications in heuristic search-based planners.

Section 5: Relaxed Planning Graph Heuristic
Description 5: This subsection examines the Relaxed Planning Graph heuristic, its implementation in the FF planner, and enhancements for improved guidance in heuristic search.

Section 6: Critical Path Heuristic
Description 6: This subsection details the critical path heuristic, its formulations, and its application in planning, particularly focusing on the h_m heuristic family.

Section 7: Causal Graph Heuristic
Description 7: This subsection introduces the causal graph heuristic, used in the Fast Downward planner, and describes its computation process.

Section 8: Abstraction Heuristic
Description 8: This subsection explores abstraction heuristics, explaining the concept and discussing key techniques such as pattern database heuristics.

Section 9: Landmark Heuristic
Description 9: This subsection discusses the landmark heuristic, how it is derived, and its application in planners like LAMA.

Section 10: HEURISTIC FOR OTHER PLANNING PARADIGMS
Description 10: This section discusses the application of heuristic techniques in various other planning paradigms beyond state space planning.

Section 11: CONCLUSION
Description 11: This section summarizes the survey, highlights key trends and advancements in heuristic techniques, and outlines future research directions.