Impact of precedence constraints on complexity of scheduling problems: a survey
---
paper_title: Sequencing and scheduling : algorithms and complexity

paper_content:

Sequencing and scheduling as a research area is motivated by questions that arise in production planning, in computer control, and generally in all situations in which scarce resources have to be allocated to activities over time. In this survey, we concentrate on the area of deterministic machine scheduling. We review complexity results and optimization and approximation algorithms for problems involving a single machine, parallel machines, open shops, flow shops and job shops. We also pay attention to two extensions of this area: resource-constrained project scheduling and stochastic machine scheduling.
---
paper_title: Computationally Tractable Classes of Ordered Sets

paper_content:

Ordered sets have recently gained much importance in many applied and theoretical problems in computer science and operations research ranging from project planning via processor scheduling to sorting and retrieval problems. These problems involve partial orders as their basic structure, e.g. as precedence constraints in scheduling problems, or as comparability relation among the objects to be sorted or retrieved.
---
paper_title: Scheduling: Theory, Algorithms, and Systems

paper_content:

This book on scheduling covers theoretical models as well as scheduling problems in the real world. Author Michael Pinedo also includes a CD that contains slide-shows from industry and movies dealing with implementations of scheduling systems. The book consists of three parts. The first part focuses on deterministic scheduling with the associated combinatorial problems. The second part covers probabilistic scheduling models. In this part it is assumed that processing times and other problem data are not known in advance. The third part deals with scheduling in practice. It covers heuristics that are popular with practitioners and discusses system design and development issues. Each chapter contains a series of computational and theoretical exercises. This book is of interest to theoreticians and practitioners alike. Graduate students in operations management, operations research, industrial engineering and computer science will find the book to be an accessible and invaluable resource. Scheduling will serve as an essential reference for professionals working on scheduling problems in manufacturing and computing environments. Michael Pinedo is the Julius Schlesinger Professor of Operations Management at New York University.
---
paper_title: Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey

paper_content:

The theory of deterministic sequencing and scheduling has expanded rapidly during the past years. In this paper we survey the state of the art with respect to optimization and approximation algorithms and interpret these in terms of computational complexity theory. Special cases considered are single machine scheduling, identical, uniform and unrelated parallel machine scheduling, and open shop, flow shop and job shop scheduling. We indicate some problems for future research and include a selective bibliography.
---
paper_title: Complexity of Scheduling under Precedence Constraints

paper_content:

Precedence constraints between jobs that have to be respected in every feasible schedule generally increase the computational complexity of a scheduling problem. Occasionally, their introduction may turn a problem that is solvable within polynomial time into an NP-complete one, for which a good algorithm is highly unlikely to exist. We illustrate the use of these concepts by extending some typical NP-completeness results and simplifying their correctness proofs for scheduling problems involving precedence constraints.
---
paper_title: Scheduling Algorithms

paper_content:

Besides scheduling problems for single and parallel machines and shop scheduling problems, this book covers advanced models involving due-dates, sequence dependent changeover times and batching. Discussion also extends to multiprocessor task scheduling and problems with multi-purpose machines. Among the methods used to solve these problems are linear programming, dynamic programming, branch-and-bound algorithms, and local search heuristics. The text goes on to summarize complexity results for different classes of deterministic scheduling problems.
---
paper_title: The Coffman-Graham Algorithm Optimally Solves UET Task Systems with Overinterval Orders

paper_content:

Scheduling of unit execution time (UET) task systems on parallel machines with minimal schedule length is known to be NP-complete. The problem is polynomially solvable for some special cases. For a fixed number of parallel machines m > 2, the complexity of the problem is still open, but the problem becomes NP-hard if m is arbitrary. In this paper we characterize a new order class that properly contains quasi-interval orders and we prove that the Coffman--Graham algorithm yields optimal schedules for this new class on any number of machines. Finally, some extensions are discussed for a larger order class and for scheduling in the presence of unit communication delays.
---
paper_title: Scheduling Interval-Ordered Tasks

paper_content:

We show that unit execution time jobs subject to a precedence constraint whose complement is chordal can be scheduled in linear time on m processors. Generalizations to arbitrary execution times are NP-complete.
---
paper_title: Sequencing Jobs to Minimize Total Weighted Completion Time Subject to Precedence Constraints

paper_content:

Suppose n jobs are to be sequenced for processing by a single machine, with the object of minimizing total weighted completion time. It is shown that the problem is NP-complete if there are arbitrary precedence constraints. However, if precedence constraints are “series parallel”, the problem can be solved in O( n log n ) time. This result generalizes previous results for the more special case of rooted trees. It is also shown how a decomposition procedure suggested by Sidney can be implemented in polynomial-bounded time. Equivalence of the sequencing problem with the optimal linear ordering problem for directed graphs is discussed.
---
paper_title: A Relation between Multiprocessor Scheduling and Linear Programming

paper_content:

The general non preemptive multiprocessor scheduling problem (NPMS) is NP-Complete, while in many specific cases, the same problem is Time-polynomial. A first connection between PMS and linear programming was established by Yannanakis, Sauer and Stone, who associated to any PMS instance some specific linear program. The main result inside this paper consists in a characterization of the partially ordered structures which allow the optimal values of any associated PMS instance to be equal to the optimal values of the corresponding linear programs.
---
paper_title: Optimal scheduling on parallel machines for a new order class

paper_content:

This paper addresses the problem of scheduling n unit length tasks on m identical machines under certain precedence constraints. The aim is to compute minimal length nonpreemptive schedules. We introduce a new order class which contains properly two rich families of precedence graphs: interval orders and a subclass of the class of series parallel orders. We present a linear time algorithm to find an optimal schedule for this new order class on any number of machines.
---
paper_title: Fractional dimension of partial orders

paper_content:

Given a partially ordered setP=(X, ≤), a collection of linear extensions {L1,L2,...,Lr} is arealizer if, for every incomparable pair of elementsx andy, we havex<y in someLi (andy<x in someLj). For a positive integerk, we call a multiset {L1,L2,...,Lt} ak-fold realizer if for every incomparable pairx andy we havex<y in at leastk of theLi's. Lett(k) be the size of a smallestk-fold realizer ofP; we define thefractional dimension ofP, denoted fdim(P), to be the limit oft(k)/k ask→∞. We prove various results about the fractional dimension of a poset.
---
paper_title: Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey

paper_content:

The theory of deterministic sequencing and scheduling has expanded rapidly during the past years. In this paper we survey the state of the art with respect to optimization and approximation algorithms and interpret these in terms of computational complexity theory. Special cases considered are single machine scheduling, identical, uniform and unrelated parallel machine scheduling, and open shop, flow shop and job shop scheduling. We indicate some problems for future research and include a selective bibliography.
---
paper_title: Scheduling Algorithms

paper_content:

Besides scheduling problems for single and parallel machines and shop scheduling problems, this book covers advanced models involving due-dates, sequence dependent changeover times and batching. Discussion also extends to multiprocessor task scheduling and problems with multi-purpose machines. Among the methods used to solve these problems are linear programming, dynamic programming, branch-and-bound algorithms, and local search heuristics. The text goes on to summarize complexity results for different classes of deterministic scheduling problems.
---
paper_title: Complexity results for scheduling chains on a single machine

paper_content:

Abstract We investigate the computational complexity of deterministic sequencing problems in which unit-time jobs have to be scheduled on a single machine subject to chain-like precedence constraints. NP-hardness is established for the cases in which the number of late jobs or the total weighted tardiness is to be minimized, and for several related problems involving the total weighted completion time criterion.
---
paper_title: Minimizing Total Tardiness on a Single Machine with Precedence Constraints

paper_content:

The problem of minimizing the total tardiness for a set of unit-processing-time jobs on a single machine is considered. J. K. Lenstra and A. H. G. Rinnooy Kan have shown that the problem is NP-hard if the jobs have arbitrary precedence constraints. They asked whether the problem remains NP-hard for tree-structured precedence constraints. In this paper we show that the problem is NP-hard even for a set of chains. Our result gives a sharp boundary for the complexity of this problem, since there is a simple, polynomial-time algorithm for a set of independent jobs. INFORMS Journal on Computing , ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.
---
paper_title: Single machine precedence constrained scheduling is a vertex cover problem

paper_content:

In this paper we study the single machine precedence constrained scheduling problem of minimizing the sum of weighted completion time. Specifically, we settle an open problem first raised by Chudak & Hochbaum and whose answer was subsequently conjectured by Correa & Schulz. ::: ::: The most significant implication of our result is that the addressed scheduling problem is a special case of the vertex cover problem. This will hopefully be an important step towards proving that the two problems behave identically in terms of approximability. ::: ::: As a consequence of our result, previous results for the scheduling problem can be explained, and in some cases improved, by means of vertex cover theory. For example, our result implies the existence of a polynomial time algorithm for the special case of two-dimensional partial orders. This considerably extends Lawler's result from 1978 for series-parallel orders.
---
paper_title: Decomposition Algorithms for Single-Machine Sequencing with Precedence Relations and Deferral Costs

paper_content:

A one-machine deterministic job-shop sequencing problem is considered. Associated with each job is its processing time and linear deferral cost. In addition, the jobs are related by a general precedence relation. The objective is to order the jobs so as to minimize the sum of the deferral costs, subject to the constraint that the ordering must be consistent with the precedence relation. A decomposition algorithm is presented, and it is proved that a permutation is optimal if and only if it can be generated by this algorithm. Four special network structures are then considered, and specializations of the general algorithm are presented.
---
paper_title: Single-machine scheduling with deteriorating jobs under a series-parallel graph constraint

paper_content:

This paper considers single-machine scheduling problems with deteriorating jobs, i.e., jobs whose processing times are an increasing function of their starting times. In addition, the jobs are related by a series-parallel graph. It is shown that for the general linear problem to minimize the makespan, polynomial algorithms exist. It is also shown that for the proportional linear problem of minimization of the total weighted completion time, polynomial algorithms exist, too.
---
paper_title: Single-machine scheduling with precedence constraints and position-dependent processing times

paper_content:

Abstract In this paper we consider single-machine scheduling problems with position-dependent processing times, i.e., jobs whose processing times are an increasing or decreasing function of their positions in a processing sequence. In addition, the jobs are related by parallel chains and a series–parallel graph precedence constraints, respectively. It is shown that for the problems of minimization of the makespan polynomial algorithms exist.
---
paper_title: Sequencing Jobs to Minimize Total Weighted Completion Time Subject to Precedence Constraints

paper_content:

Suppose n jobs are to be sequenced for processing by a single machine, with the object of minimizing total weighted completion time. It is shown that the problem is NP-complete if there are arbitrary precedence constraints. However, if precedence constraints are “series parallel”, the problem can be solved in O( n log n ) time. This result generalizes previous results for the more special case of rooted trees. It is also shown how a decomposition procedure suggested by Sidney can be implemented in polynomial-bounded time. Equivalence of the sequencing problem with the optimal linear ordering problem for directed graphs is discussed.
---
paper_title: Sequencing with Series-Parallel Precedence Constraints

paper_content:

One of the most important ideas in the theory of sequencing and scheduling is the method of adjacent pairwise job interchange. This method compares the costs of two sequences which differ only by interchanging a pair of adjacent jobs. In 1956, W. E. Smith defined a class of problems for which a total preference ordering of the jobs exists with the property that in any sequence, whenever two adjacent jobs are not in preference order, they may be interchanged with no resultant cost increase. In such a case the unconstrained sequencing problem is easily solved by sequencing the jobs in preference order. ::: ::: In this paper, a natural subclass of these problems is considered for which such a total preference ordering exists for all subsequences of jobs. The main result is an efficient general algorithm for these sequencing problems with series-parallel precedence constraints. These problems include the least cost fault detection problem, the one-machine total weighted completion time problem, the two-machine maximum completion time flow-shop problem and the maximum cumulative cost problem.
---
paper_title: Single machine scheduling models with deterioration and learning: handling precedence constraints via priority generation

paper_content:

We consider various single machine scheduling problems in which the processing time of a job depends either on its position in a processing sequence or on its start time. We focus on problems of minimizing the makespan or the sum of (weighted) completion times of the jobs. In many situations we show that the objective function is priority-generating, and therefore the corresponding scheduling problem under series-parallel precedence constraints is polynomially solvable. In other situations we provide counter-examples that show that the objective function is not priority-generating.
---
paper_title: On the Approximability of Single-Machine Scheduling with Precedence Constraints

paper_content:

We consider the single-machine scheduling problem to minimize the weighted sum of completion times under precedence constraints. In a series of recent papers, it was established that this scheduling problem is a special case of minimum weighted vertex cover. ::: ::: In this paper, we show that the vertex cover graph associated with the scheduling problem is exactly the graph of incomparable pairs defined in the dimension theory of partial orders. Exploiting this relationship allows us to present a framework for obtaining (2-2/f)-approximation algorithms, provided that the set of precedence constraints has fractional dimension of at most f. Our approach yields the best-known approximation ratios for all previously considered special classes of precedence constraints, and it provides the first results for bounded degree and orders of interval dimension 2. ::: ::: On the negative side, we show that the addressed problem remains NP-hard even when restricted to the special case of interval orders. Furthermore, we prove that the general problem, if a fixed cost present in all feasible schedules is ignored, becomes as hard to approximate as vertex cover. We conclude by giving the first inapproximability result for this problem, showing under a widely believed assumption that it does not admit a polynomial-time approximation scheme.
---
paper_title: Scheduling Opposing Forests

paper_content:

A basic problem of deterministic scheduling theory is that of scheduling n unit-length tasks on m identical processors subject to precedence constraints so as to meet a given overall deadline. T. C. Hu’s classic “level algorithm” can be used to solve this problem in linear time if the precedence constraints have the form of an in-forest or an out-forest. We show that a polynomial time algorithm for a wider class of precedence constraints is unlikely, by proving the problem to be NP-complete for precedence constraints that are the disjoint union of an in-forest and an out-forest (the “opposing forests” of our title). However, for any fixed value of m we show that this problem can be solved in polynomial time for such precedence constraints. For the special case of $m = 3$ we provide a linear time algorithm.
---
paper_title: Complexity of Scheduling under Precedence Constraints

paper_content:

Precedence constraints between jobs that have to be respected in every feasible schedule generally increase the computational complexity of a scheduling problem. Occasionally, their introduction may turn a problem that is solvable within polynomial time into an NP-complete one, for which a good algorithm is highly unlikely to exist. We illustrate the use of these concepts by extending some typical NP-completeness results and simplifying their correctness proofs for scheduling problems involving precedence constraints.
---
paper_title: Sequencing Jobs to Minimize Total Weighted Completion Time Subject to Precedence Constraints

paper_content:

Suppose n jobs are to be sequenced for processing by a single machine, with the object of minimizing total weighted completion time. It is shown that the problem is NP-complete if there are arbitrary precedence constraints. However, if precedence constraints are “series parallel”, the problem can be solved in O( n log n ) time. This result generalizes previous results for the more special case of rooted trees. It is also shown how a decomposition procedure suggested by Sidney can be implemented in polynomial-bounded time. Equivalence of the sequencing problem with the optimal linear ordering problem for directed graphs is discussed.
---
paper_title: Complexity of Scheduling under Precedence Constraints

paper_content:

Precedence constraints between jobs that have to be respected in every feasible schedule generally increase the computational complexity of a scheduling problem. Occasionally, their introduction may turn a problem that is solvable within polynomial time into an NP-complete one, for which a good algorithm is highly unlikely to exist. We illustrate the use of these concepts by extending some typical NP-completeness results and simplifying their correctness proofs for scheduling problems involving precedence constraints.
---
paper_title: Complexity of machine scheduling problems

paper_content:

We survey and extend the results on the complexity of machine scheduling problems. After a brief review of the central concept of NP-completeness we give a classification of scheduling problems on single, different and identical machines and study the influence of various parameters on their complexity. The problems for which a polynomial-bounded algorithm is available are listed and NP-completeness is established for a large number of other machine scheduling problems. We finally discuss some questions that remain unanswered.
---
paper_title: Scheduling chain-structured tasks to minimize makespan and mean flow time

paper_content:

We consider the problem of scheduling a set of chains onm > 1 identical processors with the objectives of minimizing the makespan and the mean flow time. We show that finding a nonpreemptive schedule with the minimum makespan is strongly NP-hard for each fixedm > 1, answering the open question of whether this problem is strongly NP-hard for trees. We also show that finding a nonpreemptive schedule with the minimum mean flow time is strongly NP-hard for each fixedm > 1, improving the known strong NP-hardness results for in-trees and out-trees. Finally, we generalize the result of McNaughton, showing that preemption cannot reduce the mean weighted flow time for a set of chains. The last two results together imply that finding a preemptive schedule with the minimum mean flow time is also strongly NP-hard for each fixedm > 1, answering another open question on the complexity of this problem for trees.
---
paper_title: Optimal scheduling for two-processor systems

paper_content:

Despite the recognized potential of multiprocessing little is known concerning the general problem of finding efficient algorithms which compute minimallength schedules for given computations and m?2 processors. In this paper we formulate a general model of computation structures and exhibit an efficient algorithm for finding optimal nonpreemptive schedules for these structures on two-processor systems. We prove that the algorithm gives optimal solutions and discuss its application to preemptive scheduling disciplines.
---
paper_title: Optimality of HLF for scheduling divide-and-conquer UET task graphs on identical parallel processors

paper_content:

The problem of scheduling a set of n unit execution time (UET) tasks subject to precedence constraints on m identical parallel processors is known to be NP-hard in the strong sense. However, polynomial time algorithms exist for some classes of precedence graphs. In this paper, we consider a class of divide-and-conquer graphs that naturally models the execution of the recursive control abstraction of divide-and-conquer algorithms. We prove that the Highest Level First (HLF) strategy minimizes the schedule length for this class, thus settling a conjecture of Rayward-Smith and Clark.
---
paper_title: Linear-Time Algorithms for Scheduling on Parallel Processors

paper_content:

Linear-time algorithms are presented for several problems of scheduling n equal-length tasks on m identical parallel processors subject to precedence constraints. This improves upon previous time bounds for the maximum lateness problem with treelike precedence constraints, the number-of-late-tasks problem without precedence constraints, and the one machine maximum lateness problem with general precedence constraints.
---
paper_title: Scheduling Opposing Forests

paper_content:

A basic problem of deterministic scheduling theory is that of scheduling n unit-length tasks on m identical processors subject to precedence constraints so as to meet a given overall deadline. T. C. Hu’s classic “level algorithm” can be used to solve this problem in linear time if the precedence constraints have the form of an in-forest or an out-forest. We show that a polynomial time algorithm for a wider class of precedence constraints is unlikely, by proving the problem to be NP-complete for precedence constraints that are the disjoint union of an in-forest and an out-forest (the “opposing forests” of our title). However, for any fixed value of m we show that this problem can be solved in polynomial time for such precedence constraints. For the special case of $m = 3$ we provide a linear time algorithm.
---
paper_title: Parallel Sequencing and Assembly Line Problems

paper_content:

This paper deals with a new sequencing problem in which n jobs with ordering restrictions have to be done by men of equal ability. Assume every man can do any of the n jobs. The two questions considered in this paper are 1 How to arrange a schedule that requires the minimum number of men so that all jobs are completed within a prescribed time T, and 2 if m men are available, arrange a schedule that completes all jobs at the earliest time.
---
paper_title: The Coffman-Graham Algorithm Optimally Solves UET Task Systems with Overinterval Orders

paper_content:

Scheduling of unit execution time (UET) task systems on parallel machines with minimal schedule length is known to be NP-complete. The problem is polynomially solvable for some special cases. For a fixed number of parallel machines m > 2, the complexity of the problem is still open, but the problem becomes NP-hard if m is arbitrary. In this paper we characterize a new order class that properly contains quasi-interval orders and we prove that the Coffman--Graham algorithm yields optimal schedules for this new class on any number of machines. Finally, some extensions are discussed for a larger order class and for scheduling in the presence of unit communication delays.
---
paper_title: Scheduling Interval-Ordered Tasks

paper_content:

We show that unit execution time jobs subject to a precedence constraint whose complement is chordal can be scheduled in linear time on m processors. Generalizations to arbitrary execution times are NP-complete.
---
paper_title: Optimal scheduling on parallel machines for a new order class

paper_content:

This paper addresses the problem of scheduling n unit length tasks on m identical machines under certain precedence constraints. The aim is to compute minimal length nonpreemptive schedules. We introduce a new order class which contains properly two rich families of precedence graphs: interval orders and a subclass of the class of series parallel orders. We present a linear time algorithm to find an optimal schedule for this new order class on any number of machines.
---
paper_title: Optimal scheduling for two-processor systems

paper_content:

Despite the recognized potential of multiprocessing little is known concerning the general problem of finding efficient algorithms which compute minimallength schedules for given computations and m?2 processors. In this paper we formulate a general model of computation structures and exhibit an efficient algorithm for finding optimal nonpreemptive schedules for these structures on two-processor systems. We prove that the algorithm gives optimal solutions and discuss its application to preemptive scheduling disciplines.
---
paper_title: Scheduling precedence graphs of bounded height

paper_content:

The existence of a schedule for a partially ordered set of unit length tasks on m identical processors is known to be NP-complete (J. D. Ullman, NP-complete scheduling problems, J. Comput. System Sci., 10 (1975), 384–393). The problem remains NP-complete even if we restrict the precedence graph to be of height bounded by a constant. (J. K. Lenkstra and A. H. G. Rinnooy Kan, Complexity of scheduling under precedence constraints, Operations Res., 26 (1978), 22–35; D. Dolev and M. K. Warmuth, “Scheduling Flat Graphs,” IBM Research Report RJ 3398, 1982). In these NP-completeness proofs the upper bound on the number of available processors varies with the problem instance. We present a polynomial algorithm for the case where the upper bound on the number of available processors and the height of the precedence graph are both constants.
---
paper_title: Profile Scheduling of Opposing Forests and Level Orders

paper_content:

The question of existence of a schedule of a given length for n unit length tasks on n identical processors subject to precedence constraints is known to be NP-complete [Ullman, J. Comput. System Sci., 10 (1976), pp. 384–393]. For a fixed value of m we present polynomial algorithms to find an optimal schedule for two families of precedence graphs: level orders and opposing forests. In the case of opposing forest our algorithm is a considerable improvement over the algorithm presented in [Garey et al., SIAM J. Alg. Disc. Meth., 4 (1983), pp. 72–93].
---
paper_title: On a parallel machine scheduling problem with precedence constraints

paper_content:

We characterize a nontrivial special case with a polynomial-time algorithm for a well-known parallel machine scheduling problem with precedence constraints, with a fixed number of machines, and with tasks of unit length. The special case is related to instances with given maximum path length and maximum degree of the task precedence graph. The method is based on the observation that the number of tasks is either small and bounded by a constant depending on the maximum path length and maximum degree, or alternatively, the number of tasks is large, giving a "dense" schedule.
---
paper_title: Scheduling Opposing Forests

paper_content:

A basic problem of deterministic scheduling theory is that of scheduling n unit-length tasks on m identical processors subject to precedence constraints so as to meet a given overall deadline. T. C. Hu’s classic “level algorithm” can be used to solve this problem in linear time if the precedence constraints have the form of an in-forest or an out-forest. We show that a polynomial time algorithm for a wider class of precedence constraints is unlikely, by proving the problem to be NP-complete for precedence constraints that are the disjoint union of an in-forest and an out-forest (the “opposing forests” of our title). However, for any fixed value of m we show that this problem can be solved in polynomial time for such precedence constraints. For the special case of $m = 3$ we provide a linear time algorithm.
---
paper_title: Optimal scheduling for two-processor systems

paper_content:

Despite the recognized potential of multiprocessing little is known concerning the general problem of finding efficient algorithms which compute minimallength schedules for given computations and m?2 processors. In this paper we formulate a general model of computation structures and exhibit an efficient algorithm for finding optimal nonpreemptive schedules for these structures on two-processor systems. We prove that the algorithm gives optimal solutions and discuss its application to preemptive scheduling disciplines.
---
paper_title: Computationally Tractable Classes of Ordered Sets

paper_content:

Ordered sets have recently gained much importance in many applied and theoretical problems in computer science and operations research ranging from project planning via processor scheduling to sorting and retrieval problems. These problems involve partial orders as their basic structure, e.g. as precedence constraints in scheduling problems, or as comparability relation among the objects to be sorted or retrieved.
---
paper_title: Optimal scheduling of unit-time tasks on two uniform processors under tree-like precedence constraints

paper_content:

An O(n3/(b+1)) time algorithm to obtain a minimum finish time schedule subject to tree-like precedence constraints for unit-time tasks on two uniform processors is obtained. It is assumed that the slower processor takesb time units for each one taken by the speedier one, for some integerb. It is also noted that a slight modification of this schedule yields a minimum mean flow time schedule.
---
paper_title: Scheduling Interval-Ordered Tasks

paper_content:

We show that unit execution time jobs subject to a precedence constraint whose complement is chordal can be scheduled in linear time on m processors. Generalizations to arbitrary execution times are NP-complete.
---
paper_title: Minimizing total completion time for UET tasks with release time and outtree precedence constraints

paper_content:

Brucker et al. (Math Methods Oper Res 56: 407–412, 2003) have given an O(n 2 )-time algorithm for the problems $$P \mid p_{j}=1, r_{j}$$ , outtree $$\mid \sum C_{j}$$ and $$P \mid pmtn, p_{j}=1, r_{j}$$ , outtree $$\mid \sum C_{j}$$ . In this note, we show that their algorithm admits an O(n log n)-time implementation. Copyright Springer-Verlag 2005
---
paper_title: Scheduling identical jobs with chain precedence constraints on two uniform machines

paper_content:

The problem of scheduling identical jobs with chain precedence constraints on two uniform machines is considered. It is shown that the corresponding makespan problem can be solved in linear time.
---
paper_title: Minimizing mean flow time for UET tasks

paper_content:

We consider the problem of scheduling a set of n unit-execution-time (UET) tasks, with precedence constraints, on m ≥ 1 parallel and identical processors so as to minimize the mean flow time. For two processors, the Coffman--Graham algorithm gives a schedule that simultaneously minimizes the mean flow time and the makespan. The problem becomes strongly NP-hard for an arbitrary number of processors, although the complexity is not known for each fixed m ≥ 3. For arbitrary precedence constraints, we show that the Coffman--Graham algorithm gives a schedule with a worst-case bound no more than 2, and we give examples showing that the bound is tight. For intrees, the problem can be solved in polynomial time for each fixed m ≥ 1, although the complexity is not known for an arbitrary number of processors. We show that Hu's algorithm (which is optimal for the makespan objective) yields a schedule with a worst-case bound no more than 1.5, and we give examples showing that the ratio can approach 1.308999.
---
paper_title: Preemptive scheduling of precedence-constrained jobs on parallel machines : (preprint)

paper_content:

Polynomial time-bounded algorithms are presented for solving three problems involving the preemptive scheduling of precedence-constrained jobs on parallel machines: the “intree problem”, the “two-machine problem with equal release dates”, and the “general two-machine problem”. These problems are preemptive counterparts of problems involving the nonpreemptive scheduling of unit-time jobs previously solved by Brucker, Garey and Johnson and by Garey and Johnson. The algorithms and proofs (and the running times of the algorithms) closely parallel those presented in their papers. These results improve on previous results in preemptive scheduling and also suggest a close relationship between preemptive scheduling problems and problems in nonpreemptive scheduling of unit-time jobs.
---
paper_title: Parallel Sequencing and Assembly Line Problems

paper_content:

This paper deals with a new sequencing problem in which n jobs with ordering restrictions have to be done by men of equal ability. Assume every man can do any of the n jobs. The two questions considered in this paper are 1 How to arrange a schedule that requires the minimum number of men so that all jobs are completed within a prescribed time T, and 2 if m men are available, arrange a schedule that completes all jobs at the earliest time.
---
paper_title: On preemption redundancy in scheduling unit processing time jobs on two parallel machines

paper_content:

McNaughton’s theorem (1959) states that preemptions in scheduling arbitrary processing time jobs on identical parallel machines to minimize the total weighted completion time are redundant. Du, Leung and Young (1991) proved that this remains true even though the jobs have precedence constraints in the form of chains. There are known simple counterexamples showing that other extensions of McNaughton’s theorem to other criteria or more general precedence constraints such as intrees or outtrees, or different release dates of jobs, or different speeds of machines, are not true even for equal weights of jobs. In this paper we show that in the case of two machines and unit processing times, preemptions are still advantageous for intrees or machines with different speeds even for equal weights, or outtrees for different weights, but become redundant for outtrees and equal weights even for different release dates. We also conjecture that the latter statement is actually true for any number of machines.
---
paper_title: Preemptive Scheduling of Interval Orders is Polynomial

paper_content:

In 1979, Papadimitriou and Yannakakis gave a polynomial time algorithm for the scheduling of jobs requiring unit completion times when the precedence constraints form an interval order. The authors solve here the corresponding problem, for preemptive scheduling (a job can be interrupted to work on more important tasks, and completed at a later time, subject to the usual scheduling constraints.) The m-machine preemptive scheduling problem is shown to have a polynomial algorithm, for both unit time and variable execution times as well, when the precedence constraints are given by an interval order.
---
paper_title: Optimal preemptive scheduling on a fixed number of identical parallel machines

paper_content:

In this paper, we consider the preemptive scheduling problem on a fixed number of identical parallel machines. We present a polynomial-time algorithm for finding a minimal length schedule for an order class which contains properly interval orders.
---
paper_title: Minimizing total completion time for UET tasks with release time and outtree precedence constraints

paper_content:

Brucker et al. (Math Methods Oper Res 56: 407–412, 2003) have given an O(n 2 )-time algorithm for the problems $$P \mid p_{j}=1, r_{j}$$ , outtree $$\mid \sum C_{j}$$ and $$P \mid pmtn, p_{j}=1, r_{j}$$ , outtree $$\mid \sum C_{j}$$ . In this note, we show that their algorithm admits an O(n log n)-time implementation. Copyright Springer-Verlag 2005
---
paper_title: Scheduling chain-structured tasks to minimize makespan and mean flow time

paper_content:

We consider the problem of scheduling a set of chains onm > 1 identical processors with the objectives of minimizing the makespan and the mean flow time. We show that finding a nonpreemptive schedule with the minimum makespan is strongly NP-hard for each fixedm > 1, answering the open question of whether this problem is strongly NP-hard for trees. We also show that finding a nonpreemptive schedule with the minimum mean flow time is strongly NP-hard for each fixedm > 1, improving the known strong NP-hardness results for in-trees and out-trees. Finally, we generalize the result of McNaughton, showing that preemption cannot reduce the mean weighted flow time for a set of chains. The last two results together imply that finding a preemptive schedule with the minimum mean flow time is also strongly NP-hard for each fixedm > 1, answering another open question on the complexity of this problem for trees.
---
paper_title: Scheduling preemptive jobs with precedence constraints on parallel machines

paper_content:

Abstract In parallel machine scheduling problems with general precedence constraints, it is advantageous to interrupt jobs and complete them at a later time in order to minimize Makespan. For general precedence constraints, we know that this problem is in the class of NP-hard combinatorial problems for which the development of efficient algorithms is unlikely. In this paper we propose two new heuristics SUPIO and INFIO which, respectively, give an upper bound and a lower bound to Makespan. We show numerically that the ratio between these two bounds is very close to one. SUPIO heuristic provides an optimal solution in the following cases: P2 | ,pmnt| C max , P| forest,pmnt| C max and P| interval order,pmnt| C max . Its absolute performance ratio is upper bounded by (2 − 2/ m ). We illustrate the efficiency of our heuristics by computational results associated with randomly generated problems.We show in the conclusion that this new approach could solve some non-preemptive parallel machine scheduling problems.
---
paper_title: Two machine preemptive scheduling problem with release dates, equal processing times and precedence constraints

paper_content:

Abstract We consider a scheduling problem with two identical parallel machines and n jobs. For each job we are given its release date when job becomes available for processing. All jobs have equal processing times. Preemptions are allowed. There are precedence constraints between jobs which are given by a (di)graph consisting of a set of outtrees and a number of isolated vertices. The objective is to find a schedule minimizing mean flow time. We suggest an O( n 2 ) algorithm to solve this problem. The suggested algorithm also can be used to solve the related two-machine open shop problem with integer release dates, unit processing times and analogous precedence constraints.
---
paper_title: Digraph width measures in parameterized algorithmics

paper_content:

Abstract In contrast to undirected width measures such as tree-width, which have provided many important algorithmic applications, analogous measures for digraphs such as directed tree-width or DAG-width do not seem so successful. Several recent papers have given some evidence on the negative side. We confirm and consolidate this overall picture by thoroughly and exhaustively studying the complexity of a range of directed problems with respect to various parameters, and by showing that they often remain NP-hard even on graph classes that are restricted very beyond having small DAG-width. On the positive side, it turns out that clique-width (of digraphs) performs much better on virtually all considered problems, from the parameterized complexity point of view.
---
paper_title: Scheduling and Fixed-Parameter Tractability

paper_content:

Fixed-parameter tractability analysis and scheduling are two core domains of combinatorial optimization which led to deep understanding of many important algorithmic questions. However, even though fixed-parameter algorithms are appealing for many reasons, no such algorithms are known for many fundamental scheduling problems.
---
paper_title: Graph minors. II. Algorithmic aspects of tree-width

paper_content:

We introduce an invariant of graphs called the tree-width, and use it to obtain a polynomially bounded algorithm to test if a graph has a subgraph contractible to H, where H is any fixed planar graph. We also nonconstructively prove the existence of a polynomial algorithm to test if a graph has tree-width ≤ w, for fixed w. Neither of these is a practical algorithm, as the exponents of the polynomials are large. Both algorithms are derived from a polynomial algorithm for the DISJOINT CONNECTING PATHS problem (with the number of paths fixed), for graphs of bounded tree-width.
---
paper_title: On the Approximability of Single-Machine Scheduling with Precedence Constraints

paper_content:

We consider the single-machine scheduling problem to minimize the weighted sum of completion times under precedence constraints. In a series of recent papers, it was established that this scheduling problem is a special case of minimum weighted vertex cover. ::: ::: In this paper, we show that the vertex cover graph associated with the scheduling problem is exactly the graph of incomparable pairs defined in the dimension theory of partial orders. Exploiting this relationship allows us to present a framework for obtaining (2-2/f)-approximation algorithms, provided that the set of precedence constraints has fractional dimension of at most f. Our approach yields the best-known approximation ratios for all previously considered special classes of precedence constraints, and it provides the first results for bounded degree and orders of interval dimension 2. ::: ::: On the negative side, we show that the addressed problem remains NP-hard even when restricted to the special case of interval orders. Furthermore, we prove that the general problem, if a fixed cost present in all feasible schedules is ignored, becomes as hard to approximate as vertex cover. We conclude by giving the first inapproximability result for this problem, showing under a widely believed assumption that it does not admit a polynomial-time approximation scheme.
---
paper_title: Optimal scheduling for two-processor systems

paper_content:

Despite the recognized potential of multiprocessing little is known concerning the general problem of finding efficient algorithms which compute minimallength schedules for given computations and m?2 processors. In this paper we formulate a general model of computation structures and exhibit an efficient algorithm for finding optimal nonpreemptive schedules for these structures on two-processor systems. We prove that the algorithm gives optimal solutions and discuss its application to preemptive scheduling disciplines.
---
paper_title: Complexity results for scheduling chains on a single machine

paper_content:

Abstract We investigate the computational complexity of deterministic sequencing problems in which unit-time jobs have to be scheduled on a single machine subject to chain-like precedence constraints. NP-hardness is established for the cases in which the number of late jobs or the total weighted tardiness is to be minimized, and for several related problems involving the total weighted completion time criterion.
---
paper_title: Single-machine scheduling with deteriorating jobs under a series-parallel graph constraint

paper_content:

This paper considers single-machine scheduling problems with deteriorating jobs, i.e., jobs whose processing times are an increasing function of their starting times. In addition, the jobs are related by a series-parallel graph. It is shown that for the general linear problem to minimize the makespan, polynomial algorithms exist. It is also shown that for the proportional linear problem of minimization of the total weighted completion time, polynomial algorithms exist, too.
---
paper_title: Complexity of machine scheduling problems

paper_content:

We survey and extend the results on the complexity of machine scheduling problems. After a brief review of the central concept of NP-completeness we give a classification of scheduling problems on single, different and identical machines and study the influence of various parameters on their complexity. The problems for which a polynomial-bounded algorithm is available are listed and NP-completeness is established for a large number of other machine scheduling problems. We finally discuss some questions that remain unanswered.
---
paper_title: Scheduling Opposing Forests

paper_content:

A basic problem of deterministic scheduling theory is that of scheduling n unit-length tasks on m identical processors subject to precedence constraints so as to meet a given overall deadline. T. C. Hu’s classic “level algorithm” can be used to solve this problem in linear time if the precedence constraints have the form of an in-forest or an out-forest. We show that a polynomial time algorithm for a wider class of precedence constraints is unlikely, by proving the problem to be NP-complete for precedence constraints that are the disjoint union of an in-forest and an out-forest (the “opposing forests” of our title). However, for any fixed value of m we show that this problem can be solved in polynomial time for such precedence constraints. For the special case of $m = 3$ we provide a linear time algorithm.
---
paper_title: Complexity of Scheduling under Precedence Constraints

paper_content:

Precedence constraints between jobs that have to be respected in every feasible schedule generally increase the computational complexity of a scheduling problem. Occasionally, their introduction may turn a problem that is solvable within polynomial time into an NP-complete one, for which a good algorithm is highly unlikely to exist. We illustrate the use of these concepts by extending some typical NP-completeness results and simplifying their correctness proofs for scheduling problems involving precedence constraints.
---
paper_title: Single-machine scheduling with precedence constraints and position-dependent processing times

paper_content:

Abstract In this paper we consider single-machine scheduling problems with position-dependent processing times, i.e., jobs whose processing times are an increasing or decreasing function of their positions in a processing sequence. In addition, the jobs are related by parallel chains and a series–parallel graph precedence constraints, respectively. It is shown that for the problems of minimization of the makespan polynomial algorithms exist.
---
paper_title: Sequencing Jobs to Minimize Total Weighted Completion Time Subject to Precedence Constraints

paper_content:

Suppose n jobs are to be sequenced for processing by a single machine, with the object of minimizing total weighted completion time. It is shown that the problem is NP-complete if there are arbitrary precedence constraints. However, if precedence constraints are “series parallel”, the problem can be solved in O( n log n ) time. This result generalizes previous results for the more special case of rooted trees. It is also shown how a decomposition procedure suggested by Sidney can be implemented in polynomial-bounded time. Equivalence of the sequencing problem with the optimal linear ordering problem for directed graphs is discussed.
---
paper_title: Single machine scheduling models with deterioration and learning: handling precedence constraints via priority generation

paper_content:

We consider various single machine scheduling problems in which the processing time of a job depends either on its position in a processing sequence or on its start time. We focus on problems of minimizing the makespan or the sum of (weighted) completion times of the jobs. In many situations we show that the objective function is priority-generating, and therefore the corresponding scheduling problem under series-parallel precedence constraints is polynomially solvable. In other situations we provide counter-examples that show that the objective function is not priority-generating.
---

Title: Impact of precedence constraints on complexity of scheduling problems: a survey

Section 1: Introduction
Description 1: Introduce the role of partial order sets in scheduling theory and the focus of the survey on complexity results according to the structure of precedence constraints.

Section 2: Some already studied orders
Description 2: Introduce the definition of various classes of partial orders that have already been studied in scheduling theory, including graph theory definitions and specific structures like intrees and series-parallel graphs.

Section 3: Scheduling notations
Description 3: Explain the standard notations used in the paper for describing scheduling problems, including the α|β|γ format.

Section 4: One-machine problems
Description 4: Focus on the impact of precedence constraints on the complexity of one-machine scheduling problems, discussing polynomial cases, NP-complete cases, and open problems.

Section 5: Parallel machines without preemption
Description 5: Discuss the complexity of scheduling problems on parallel machines without preemption, focusing on the makespan criterion for both arbitrary and fixed numbers of machines, and other criteria like total flow time.

Section 6: Parallel machines with preemption
Description 6: Examine the impact of preemption on the complexity of scheduling problems on parallel machines, discussing both makespan and total flow time criteria.

Section 7: Conclusion
Description 7: Summarize the survey, highlighting key findings, remaining open problems, and possible future research directions in the field of scheduling problems with precedence constraints.