. The satisfiability (SAT) problem is a core problem in mathematical logic and computing theory. In practice, SAT is fundamental in solving many problems in automated reasoning, computer-aided design, computeraided manufacturing, machine vision, database, robotics, integrated circuit design, computer architecture design, and computer network design. Traditional methods treat SAT as a discrete, constrained decision problem. In recent years, many optimization methods, parallel algorithms, and practical techniques have been developed for solving SAT. In this survey, we present a general framework (an algorithm space) that integrates existing SAT algorithms into a unified perspective. We describe sequential and parallel SAT algorithms including variable splitting, resolution, local search, global optimization, mathematical programming, and practical SAT algorithms. We give performance evaluation of some existing SAT algorithms. Finally, we provide a set of practical applications of the sat...

In a distributed system or communication network tasks may need to be executed on more than one processor. For time-critical tasks, the timing constraints are typically given as end-to-end release-times and deadlines. This paper describes algorithms to schedule a class of systems where all the tasks execute on different processors in turn in the same order. This end-to-end scheduling problem is known as the flow-shop problem. We present two cases where the problem is tractable and evaluate a heuristic for the N P-hard general case. We generalize the traditional flow-shop model in two directions. First, we present an algorithm for scheduling flow shops where tasks can be serviced more than once by some processors. Second, we describe a heuristic algorithm to schedule flow shops that consist of periodic tasks. Some considerations are made about scheduling systems with more than one flow shop. 1

We consider the following fundamental scheduling problem. The input to the problem consists of n jobs and k machines. Each of the jobs is associated with a release time, a deadline, a weight, and a processing time on each of the machines. The goal is to find a schedule that maximizes the weight of jobs that meet their deadline. We give constant factor approximation algorithms for four variants of the problem, depending on the type of the machines (identical vs. unrelated), and the weight of the jobs (identical vs. arbitrary). All these variants are known to be NP-Hard, and we observe that the two variants involving unrelated machines are also MAX-SNP hard. To the best of our knowledge, these are the first approximation algorithms for such problems in the non-preemptive o -line setting. The specific results obtained are: -- For identical job weights and unrelated machines: a greedy 2-approximation algorithm. -- For identical job weights and k identical machines: the same greedy alg...

This paper surveys much of the classical and current work in the area of path problems on digraphs. After a search of more than sixty five papers that reference Warshall's algorithm, we have concluded that our work on threshold transitive closure has probably not appeared in the literature. This work does not fit easily into any of the previous axiomatic treatments of Warshall's algorithm, and it may be possible to axiomize our work to solve AND/OR path problems, thereby generalizing much of the previous work. 96

We provide a review and synthesis of polyhedral approaches to machine scheduling problems. The choice of decision variables is the prime determinant of various formulations for such problems. Constraints, such as facet inducing inequalities for corresponding polyhedra, are often needed, in addition to those just required for the validity of the initial formulation, in order to obtain useful lower bounds and structural insights. We review formulations based on time–indexed variables; on linear ordering, start time and completion time variables; on assignment and positional date variables; and on traveling salesman variables. We point out relationship between various models, and provide a number of new results, as well as simplified new proofs of known results. In particular, we emphasize the important role that supermodular polyhedra and greedy algorithms play in many formulations and we analyze the strength of the lower and upper bounds obtained from different formulations and relaxations. We discuss separation algorithms for several classes of inequalities, and their potential applicability in generating cutting planes for the practical solution of such scheduling problems. We also review some recent results on approximation algorithms based on some of these formulations.

Consider a complete directed graph in which each arc has a given length. There is a set ofjobs, each job i located at some node of the graph, with an associated processing time hi, and whose execution has to start within a prespecified time window [r;, di]. We have a single server that can move on the arcs of the graph, at unit speed, and that has to execute all of the jobs within their respective time windows. We consider the following two problems: (a) minimize the time by which all jobs are executed (traveling salesman problem) and (b) minimize the sum of the waiting times of the jobs (traveling repairman problem). We focus on the following two special cases: (a) The jobs are located on a line and (b) the number of nodes of the graph is bounded by some integer constant B. Furthermore, we consider in detail the special cases where (a) all of the processing times are 0, (b) all of the release times ri are 0, and (c) all of the deadlines di are infinite. For many of the resulting problem combinations, we settle their complexity either by establishing NP-completeness or by presenting polynomial (or pseudopolynomial) time algorithms. Finally, we derive algorithms for the case where, for any time t, the number of jobs that can be executed at that time is bounded. I.

In this paper we consider the job interval selection problem (JISP), a simple scheduling model with a rich history and numerous applications. Special cases of this problem include the so-called real-time scheduling problem (also known as the throughput maximization problem) in single and multiple machine environments. In these special cases we have to maximize the number of jobs scheduled between their release date and deadline (preemption is not allowed). Even the single machine case is NP-hard. The unrelated machines case, as well as other special cases of JISP, are MAX SNP-hard. A simple greedy algorithm gives a 2-approximation for JISP. Despite many efforts, this was the best approximation guarantee known, even for throughput maximization on a single machine. In this paper, we break this barrier and show an approximation guarantee of less than 1.582 for arbitrary instances of JISP. For some special cases, we show better results.

We consider the problem of non-preemptively scheduling periodic and sporadic task systems on one processor using inserted idle times. For periodic task systems, we prove that the decision problem of determining whether a periodic task system is schedulable for all start times with respect to the class of algorithms using inserted idle times is NP-Hard in the strong sense, even when the deadlines are equal to the periods. We then show that if there exists a polynomial time scheduling algorithm which correctly schedules a periodic task system T whenever T is feasible for all start times, then P=NP. We also prove that with respect to the same class of algorithms, the problem of determining whether there exist start times for which a periodic task system is feasible is also NP-Hard in the strong sense even when the deadlines are equal to the periods. The second part of the paper concentrates on sporadic task systems and inserted idle times. It seems reasonable to suppose that to insert idl...

. This paper surveys, analyses and establishes new polynomial-time reductions among scheduling problems that connect identical parallel machines with unit-time shops and the preemption facility with chain-like precedence constraints in equal machine environments. The reductions turn out to be unexpectedly fruitful in clarifying the complexity status of many scheduling problems that were open before. New complexity results in the paper are devoted to identical parallel machines, ow shops and open shops. 1. Introduction Recently found mass polynomial-time reductions between scheduling problems on identical parallel machines and shop scheduling problems with nonpreemptive unit processing time operations proved to be very eective in studying their complexity. The latter problems we simply call unit-time shops. Reductions of unit-time shops to identical parallel machines we call parallelizings, and the inverse reductions, i.e., of identical parallel machines to unittime shops, we c...

We present the first coupled formal and empirical analysis of the Satellite Range Scheduling application. We structure our study as a progression; we start by studying a simplified version of the problem in which only one resource is present. We show that the simplified version of the problem is equivalent to a well-known machine scheduling problem and use this result to prove that Satellite Range Scheduling is NP-complete. We also show that for the one-resource version of the problem, algorithms from the machine scheduling domain outperform a genetic algorithm previously identified as one of the best algorithms for Satellite Range Scheduling. Next, we investigate if these performance results generalize for the problem with multiple resources. We exploit two sources of data: actual request data from the U.S. Air Force Satellite Control Network (AFSCN) circa 1992 and data created by our problem generator, which is designed to produce problems similar to the ones currently solved by AFSCN. Three main results emerge from our empirical study of algorithm performance for multiple-resource problems. First, the performance results obtained for the single-resource version of the problem do not generalize: the algorithms from the machine scheduling domain perform poorly for the multiple-resource problems. Second, a simple heuristic is shown to perform well on the old problems from 1992; however it fails to scale to larger, more complex generated problems. Finally, a genetic algorithm is found to yield the best overall performance on the larger, more difficult problems produced by our generator.