It has recently been shown that local search is surprisingly good at finding satisfying assignments for certain classes of CNF formulas [24]. In this paper we demonstrate that the power of local search for satisfiability testing can be further enhanced by employinga new strategy, called "mixed random walk", for escaping from local minima. We present experimental results showing how this strategy allows us to handle formulas that are substantially larger than those that can be solved with basic local search. We also present a detailed comparison of our random walk strategy with simulated annealing. Our results show that mixed random walk is the superior strategy on several classes of computationally difficult problem instances. Finally, we present results demonstrating the effectiveness of local search with walk for solving circuit synthesis and diagnosis problems.

. 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...

We describe a two phase algorithm for MAX--SAT and weighted MAX-- SAT problems. In the first phase, we use the GSAT heuristic to find a good solution to the problem. In the second phase, we use an enumeration procedure based on the Davis--Putnam--Loveland algorithm, to find a provably optimal solution. The first heuristic stage improves the performance of the algorithm by obtaining an upper bound on the minimum number of unsatisfied clauses that can be used in pruning branches of the search tree. We compare our algorithm with an integer programming branch and cut algorithm. Our implementation of the two phase algorithm is faster Research partially supported by ONR Grant number N00014--94--1--0391. y Mathematics Department, New Mexico Tech, Socorro, NM 87801. z Department of Mathematical Sciences, Clemson University, Clemson, SC 29634 than the integer programming approach on many problems. However, the integer programming approach is more effective than the two phase algorithm o...

A new Reactive Local Search (RLS ) algorithm is proposed for the solution of the Maximum-Clique problem. RLS is based on local search complemented by a feedback (history-sensitive) scheme to determine the amount of diversification. The reaction acts on the single parameter that decides the temporary prohibition of selected moves in the neighborhood, in a manner inspired by Tabu Search. The performance obtained in computational tests appears to be significantly better with respect to all algorithms tested at the the second DIMACS implementation challenge. The worst-case complexity per iteration of the algorithm is O(max{n, m}) where n and m are the number of nodes and edges of the graph. In practice, when a vertex is moved, the number of operations tends to be proportional to its number of missing edges and therefore the iterations are particularly fast in dense graphs.

Local search algorithms are among the standard methods for solving hard combinatorial problems from various areas of Artificial Intelligence and Operations Research. For SAT, some of the most successful and powerful algorithms are based on stochastic local search and in the past 10 years a large number of such algorithms have been proposed and investigated. In this article, we focus on two particularly well-known families of local search algorithms for SAT, the GSAT and WalkSAT architectures. We present a detailed comparative analysis of these algorithms' performance using a benchmark set which contains instances from randomised distributions as well as SAT-encoded problems from various domains. We also investigate the robustness of the observed performance characteristics as algorithm-dependent and problem-dependent parameters are changed. Our empirical analysis gives a very detailed picture of the algorithms' performance for various domains of SAT problems; it also reveals a fundamental weakness in some of the best-performing algorithms and shows how this can be overcome.

Satisfiability is a class of NP-complete problems that model a wide range of real-world applications. These problems are difficult to solve because they have many local minima in their search space, often trapping greedy search methods that utilize some form of descent. In this paper, we propose a new discrete Lagrange-multiplier-based global-search method for solving satisfiability problems. We derive new approaches for applying Lagrangian methods in discrete space, show that equilibrium is reached when a feasible assignment to the original problem is found, and present heuristic algorithms to look for equilibrium points. Instead of restarting from a new starting point when a search reaches a local trap, the Lagrange multipliers in our method provide a force to lead the search out of a local minimum and move it in the direction provided by the Lagrange multipliers. One of the major advantages of our method is that it has very few algorithmic parameters to be tuned by users, and the se...

Abstract. In this paper we introduce UBCSAT, a new implementation and experimentation environment for Stochastic Local Search (SLS) algorithms for SAT and MAX-SAT. Based on a novel triggered procedure architecture, UBCSAT provides implementations of numerous well-known and widely used SLS algorithms for SAT and MAX-SAT, including GSAT, WalkSAT, and SAPS; these implementations generally match or exceed the efficiency of the respective original reference implementations. Through numerous reporting and statistical features, including the measurement of run-time distributions, UBCSAT facilitates the advanced empirical analysis of these algorithms. New algorithm variants, SLS algorithms, and reporting features can be added to UBCSAT in a straightforward and efficient way. UBCSAT is implemented in C and runs on numerous platforms and operating systems; it is publicly and freely available at

Stochastic local search is an effective technique for solving certain classes of large, hard propositional satisfiability problems, including propositional encodings of problems such as circuit synthesis and graph coloring (Selman, Levesque, and Mitchell 1992; Selman, Kautz, and Cohen 1994). Many problems of interest to AI and operations research cannot be conveniently encoded as simple satisfiability, because they involve both hard and soft constraints -- that is, any solution may have to violate some of the less important constraints. We show how both kinds of constraints can be handled by encoding problems as instances of weighted MAX-SAT (finding a model that maximizes the sum of the weights of the satisfied clauses that make up a problem instance). We generalize our local-search algorithm for satisfiability (GSAT) to handle weighted MAX-SAT, and present experimental results on encodings of the Steiner tree problem, which is a well-studied hard combinatorial search problem. On many...

For the last ten years, a significant amount of work in the constraint community has been devoted to the improvement of complete methods for solving soft constraints networks. We wanted to see how recent progress in the weighted CSP (WCSP) field could compete with other approaches in related fields.

. Many AI problems can be conveniently encoded as discrete constraint satisfaction problems. It is often the case that not all solutions to a CSP are equally desirable --- in general, one is interested in a set of "preferred" solutions (for example, solutions that minimize some cost function) . Preferences can be encoded by incorporating "soft" constraints in the problem instance. We show how both hard and soft constraints can be handled by encoding problems as instances of weighted MAX-SAT (finding a model that maximizes the sum of the weights of the satisfied clauses that make up a problem instance). We generalize a local-search algorithm for satisfiability to handle weighted MAX-SAT. To demonstrate the effectiveness of our approach, we present experimental results on encodings of a set of well-studied network Steiner-tree problems. This approach turns out to be competitive with some of the best current specialized algorithms developed in operations research. 1. Introduction Traditi...