A new algorithm for testing satisfiability of propositional formulas in conjunctive normal form (CNF) is described. It applies reasoning in the form of certain resolution operations, and identification of equivalent literals. Resolution produces growth in the size of the formula, but within a global quadratic bound; most previous methods avoid operations that produce any growth, and generally do not identify equivalent literals. Computational experience indicates that the method does substantially less "guessing" than previously reported algorithms, while keeping a polynomial time bound on the work done between guesses. Experiments indicate that, for larger problems, the time investment in reasoning returns a profit in reduced searching, and the profit increases with increasing problem size. Experimental data compares six branching strategies of the proposed algorithm on a variety of problems, including several Dimacs benchmarks. These branching strategies were shown to perform differently with statistical signi cance. A new scheme based on Johnson's maximum satisfiability approximation algorithm was found to be the best overall. Both satisfiable and unsatifi able random 3-CNF formulas with 50-283 variables and 4.27 ratio of clauses to variables have been tested; this class is generally acknowledged to be relatively "hard" and

Abstract not found

Stochastic local search (SLS) algorithms have been successfully applied to hard combinatorial problems from different domains. Due to their inherent randomness, the run-time behaviour of these algorithms is characterised by a random variable. The detailed knowledge of the run-time distribution provides important information about the behaviour of SLS algorithms. In this paper we investigate the empirical run-time distributions for Walksat, one of the most powerful SLS algorithms for the Propositional Satisfiability Problem (SAT). Using statistical analysis techniques, we show that on hard Random-3-SAT problems, Walksat's run-time behaviour can be characterised by exponential distributions. This characterisation can be generalised to various SLS algorithms for SAT and to encoded problems from other domains. This result also has a number of consequences which are of theoretical as well as practical interest. One of these is the fact that these algorithms can be easily parallelised such that optimal speed-up is achieved for hard problem instances.

Benders decomposition uses a strategy of “learning from one’s mistakes.” The aim of this paper is to extend this strategy to a much larger class of problems. The key is to generalize the linear programming dual used in the classical method to an “inference dual. ” Solution of the inference dual takes the form of a logical deduction that yields Benders cuts. The dual is therefore very different from other generalized duals that have been proposed. The approach is illustrated by working out the details for propositional satisfiability and 0-1 programming problems. Computational tests are carried out for the latter, but the most promising contribution of logic-based Benders may be to provide a framework for combining optimization and constraint programming methods.

Phase transitions in constraint satisfaction problems (CSP's) are the subject of intense study. We identify an order parameter for random binary CSP's. There is a rapid transition in the probability of a CSP having a solution at a critical value of this parameter. The order parameter allows different phase transition behaviour to be compared in an uniform manner, for example CSP's generated under different regimes. We then show that within classes, the scaling of behaviour can be modelled by a tehnique called "finite size scaling". This applies not only to probability of solubility, as has been observed before in other NP-problems, but also to search cost, the first time this has been observed. Furthermore, the technique applies with equal validity to several different methods of varying problem size. As well as contributing to the understanding of phase transitions, we contribute by allowing much finer grained comparison of algorithms, and for accurate empirical extrapolations of beha...

Stochastic local search (SLS) algorithms have been successfully applied to hard combinatorial problems from different domains. One important feature of SLS algorithms is the fact that their run-time behavior is characterized by a random variable. Consequently, the detailed knowledge of the run-time distribution provides important information for the analysis of SLS algorithms. In this paper we investigate the empirical run-time distributions for several state-of-the-art stochastic local search algorithms for SAT and CSP. Using statistical analysis techniques, we show that on a variety of problems from both randomized distributions and encodings of the blocks world planning and graph coloring domains, the observed run-time behavior can be characterized by exponential distributions. As a first direct consequence of this result, we establish that these algorithms can be easily parallelized with optimal speedup.

Heuristic optimization algorithms seek good feasible solutions to optimization problems in circumstances where the complexities of the problem or the limited time available for solution do not allow exact solution. Although worst case and probabilistic analysis of algorithms have produced insight on some classic models, most of the heuristics developed for large optimization problem must be evaluated empirically—by applying procedures to a collection of specific instances and comparing the observed solution quality and computational burden. This paper focuses on the methodological issues that must be confronted by researchers undertaking such experimental evaluations of heuristics, including experimental design, sources of test instances, measures of algorithmic performance, analysis of results, and presentation in papers and talks. The questions are difficult, and there are no clear right answers. We seek only to highlight the main issues, present alternative ways of addressing them under different circumstances, and caution about pitfalls to avoid. Key Words: Heuristic optimization, computational experiments 1.

Carrying out empirical studies is widely held to be of importance. A view less widely held is that experiments should be replicated externally to verify and validate the original results. This paper serves two main functions. First, the need for external replications is established. The role of replication in experimental software engineering is discussed. Without the confirming power of external replications, results in experimental software engineering should only be provisionally accepted, if at all. An extension to the framework for experimentation in software engineering by Basili et al [5] is proposed to differentiate between the various kinds of internal and external replication and their powers of confirmation and to allow a better appreciation of the context of a piece of empirical work. Second, this paper presents a concrete example of an external replication of an experiment which tested the benefits to maintenance of using modular code against non-modular (monolithic) code....

SAT Competition 2002 held in March--May 2002 in conjunction with SAT 2002 (the Fifth International Symposium on the Theory and Applications of Satisfiability Testing). About 30 solvers and 2300 benchmarks took part in the competition, which required more than 2 CPU years to complete the evaluation. In this report

We give some dos and don'ts for those analysing algorithms experimentally. We illustrate these with many examples from our own research on the study of algorithms for NP-complete problems such as satisfiability and constraint satisfaction. Where we have not followed these maxims, we have suffered as a result. 1 Introduction The empirical study of algorithms is a relatively immature field with many technical and scientific problems. We support the calls of McGeoch (1986,1996), Hooker (1994), and others for a more scientific approach to the empirical study of algorithms. Our contribution in this paper is colloquial. We admit to a large number of mistakes in conducting our research. While painful, we hope that this will encourage others to avoid these mistakes, and thereby to develop practices which represent good science. Much of our research has been on the experimental analysis of algorithms and phase transitions in NP-complete problems, most commonly in satisfiability or constraint s...