SAT Algorithms Research Articles

GASAT: A Genetic Local Search Algorithm for the Satisfiability Problem

This paper presents GASAT, a hybrid algorithm for the satisfiability problem (SAT). The main feature of GASAT is that it includes a recombination stage based on a specific crossover and a tabu search stage. We have conducted experiments to evaluate the different components of GASAT and to compare its overall performance with state-of-the-art SAT algorithms. These experiments show that GASAT provides very competitive results.

Many hard examples in exact phase transitions

This paper analyzes the resolution complexity of two random constraint satisfaction problem (CSP) models (i.e. Model RB/RD) for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, it is proved that almost all instances of Model RB/RD have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of CSPs and CNF formulas hard to solve, which can be useful in the experimental evaluation of CSP and SAT algorithms, but also propose models with both many hard instances and exact phase transitions. Finally, conclusions are presented, as well as a detailed comparison of Model RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively, exhibit three different kinds of phase transition behavior in NP-complete problems.

On quantum versions of record-breaking algorithms for SAT

It is well known that a straightforward application of Grover's quantum search algorithm enables to solve SAT in O (2 n /2 ) steps. Ambainis (SIGACT News, 2004) observed that it is possible to use Grover's technique to similarly speed up a sophisticated algorithm for solving 3-SAT. In this note, we show that a similar speed up can be obtained for all major record-breaking algorithms for satisfiability. We also show that if we use Grover's technique only, then we cannot do better than quadratic speed up.

A Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length

We give a randomized algorithm for testing satisfiability of Boolean formulas in conjunctive normal form with no restriction on clause length. This algorithm uses the clause-shortening approach proposed by Schuler. The running time of the algorithm is O(2^(n(1−1/α))) where α = ln(m/n) + O(ln ln m) and n, m are respectively the number of variables and the number of clauses in the input formula. This bound is asymptotically better than the previously best known 2^(n(1−1/ log(2m))) bound for SAT.

UnitWalk: A New SAT Solver that Uses Local Search Guided by Unit Clause Elimination

In this paper we present a new randomized algorithm for SAT, i.e., the satisfiability problem for Boolean formulas in conjunctive normal form. Despite its simplicity, this algorithm performs well on many common benchmarks ranging from graph coloring problems to microprocessor verification. Our algorithm is inspired by two randomized algorithms having the best current worst-case upper bounds ([27,28] and [30,31]). We combine the main ideas of these algorithms in one algorithm. The two approaches we use are local search (which is used in many SAT algorithms, e.g., in GSAT [34] and WalkSAT [33]) and unit clause elimination (which is rarely used in local search algorithms). In this paper we do not prove any theoretical bounds. However, we present encouraging results of computational experiments comparing several implementations of our algorithm with other SAT solvers. We also prove that our algorithm is probabilistically approximately complete (PAC).

Incomplete Dynamic Backtracking for Linear Pseudo-Boolean Problems

Many combinatorial problems can be modeled as 0/1 integer linear programs. Problems ex- pressed in this form are usually solved by Operations Research algorithms, but good results have also been obtained using generalised SAT algorithms based on backtracking or local search, after transformation to pseudo-Boolean form. A third class of SAT algorithm uses non-systematic backtracking to combine con- straint propagation with local search-like scalability, at the cost of completeness. This paper describes such an algorithm for pseudo-Boolean models. Experimental results on a variety of problems are encouraging, in some cases yielding improved solutions or performance compared to previous algorithms.

A test sequence generation method for communication protocols using the SAT algorithm

AbstractThe specification for a communication protocol is generally represented by a finite‐state machine, and the operation of the machine is represented by transitions among states. The device implementing such specification is called the implementation under test (IUT). It is important for the IUT that its operation should be verified for all state transitions on the finite‐state machine given in the specification. This test is called the conformance testing. In such testing it is important to generate efficiently a route containing all state transitions from the initial state. This problem is called the test sequence generation problem. This paper considers the test sequence generation problem for a communication protocol and proposes an application of an algorithm for satisfiability problem (SAT) that can flexibly handle various constraints, such as the order constraint and the time constraint among multiple constraints. The proposed method is applied to the dynamic host configuration protocol (DHCP) and its effectiveness is demonstrated. © 2003 Wiley Periodicals, Inc. Syst Comp Jpn, 34(11): 20–29, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.10482

SAT problems with chains of dependent variables

This paper has two related themes. Firstly, artificial SAT problems are used to show that certain chains of variable dependency have a harmful effect on local search, sometimes causing exponential scaling on intrinsically easy problems. Secondly, systematic, local and hybrid SAT algorithms are evaluated on Hamiltonian cycle problems, exposing weaknesses in all three. The connection between the two themes is that some Hamiltonian cycle problems also cause local search to scale badly, indicating that pathological variable dependencies occur in more realistic applications. More generally, the results highlight the need for alternative models and search algorithms, and new examples of both are described.

Algorithms for Sat and Upper Bounds on Their Complexity

We survey recent algorithms for the propositional satisfiability problem. In particular, we consider algorithms having the best current worst-case upper bounds on their complexity. We also discuss some related issues: a derandomization of the algorithm of Paturi, Pudlak, Saks, and Zane, the Valiant–Vazirani lemma, and random walk algorithms with the “back button.” Bibliography: 47 titles.

The Unreasonable Effectiveness of Alternation-Based Satisfiability Algorithms

Abstract Abstract General SAT problems have associated 2SAT problems. These 2SAT problems provide support for techniques that augment existing SAT algorithms and enable new algorithms. Alternation covers yield more general termination criteria for SAT algorithms. The implication graph of the one related 2SAT problem supports decomposing SAT problems into independently solvable and deferable parts. An example demonstrates the effectiveness of the decomposition on one DIMACS benchmark problem. Another related 2SAT problem manifests constraints not recognized by other techniques. The theoretical and practical implications of these results are open.

Ensemble-based prediction of SAT search behaviour

Abstract Abstract Before attempting to solve an instance of the satisfiability problem, what can we ascertain about the instance at hand and how can we put that information to use when selecting and tuning a SAT algorithm to solve the instance? We argue for an ensemble-based approach and describe an illustrative example of how such a methodology can be applied to determine optimal restart cutoff points for systematic, backtracking search procedures for SAT. We discuss the methodology and indicate how it can be applied to evaluate such strategies as restarts, algorithm comparison, randomization and portfolios of algorithms.

Nonsystematic Search and No-Good Learning

Nonsystematic search algorithms seem, in general, to be well suited to large-scale problems with many solutions. However, they tend to perform badly for problems with few solutions, and they cannot be used for insoluble problems, since they are incomplete. Here we present a new algorithm, i>learn-SAT, that, although based on nonsystematic search, is complete. Completeness is realized through a process of no-good learning, learning-by-merging. This requires exponential space in the worst case. We show, nevertheless, that i>learn-SAT performs very well on certain SAT problems that are tightly constrained or insoluble. Indeed, its performance generally approximates the best SAT algorithms and does much better at lower clause densities. i>Learn-SAT also maintains much of the efficient performance of nonsystematic search for large-scale problems with many solutions, at least relative to backtrack search algorithms. These results indicate that the burden on memory, imposed by no-good learning, is not generally a problem for i>learn-SAT. This is perhaps surprising in view of previous work. What is even more surprising is the scalability of i>learn-SAT. For some types of problem it scales very much better than the nearest competitive algorithm. There are other types, however, for which this is not the case. The performance profile of i>learn-SAT emerges from an experimental methodology related to the one outlined by Mammen and Hogg in 1997.

Local Search Algorithms for SAT: An Empirical Evaluation

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 that contains instances from randomized 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.

The Multi-SAT algorithm

Many efficient algorithms have been developed for satisfiability testing. They complement rather than exclude each other by being effective for particular problem instances. In this research, we give a Multi-SAT algorithm for the SAT problem. The Multi-SAT algorithm integrates several efficient SAT algorithms. It makes use of different algorithmic niches for satisfiability testing. Based on cost-effective cluster computing, Multi-SAT can perform simultaneous satisfiability testing, using several “stones” to shoot one “bird”. The software architecture for Multi-SAT has been designed. A number of software tools have been developed. This software tool kit can support efficient satisfiability testing with uncertain problem structure, facilitating multiple tracking of an algorithm structure, and allow a detailed study of the entire problem spectrum. It provides a cost-effective multi-tool kit for practical satisfiability testing.

On a generalization of extended resolution

Motivated by improved SAT algorithms ((O. Kullmann, DIMACS Series, vol. 35, Amer. Math. Soc., Providence, RI, 1997; O. Kullmann, Theoret. Comput. Sci. (1999); O. Kullmann, Inform. Comput., submitted); yielding new worst-case upper bounds) a natural parameterized generalization GER of Extended Resolution (ER) is introduced. ER can simulate polynomially GER, but GER allows special cases for which exponential lower bounds can be proven.

Towards a characterisation of the behaviour of stochastic local search algorithms for SAT

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 speedup is achieved for hard problem instances.

Experimental study on strategy of combining SAT algorithms

The effectiveness of many SAT algorithms is mainly reflected by their significant performances on one or several classes of specific SAT problems. Different kinds of SAT algorithms all have their own hard instances respectively. Therefore, to get the better performance on all kinds of problems, SAT solver should know how to select different algorithms according to the feature of instances. In this paper the differences of several effective SAT algorithms are analyzed and two new parameters ϕ and δ are proposed to characterize the feature of SAT instances. Experiments are performed to study the relationship between SAT algorithms and some statistical parameters including ϕ, δ. Based on this analysis, a strategy is presented for designing a faster SAT tester by carefully combining some existing SAT algorithms. With this strategy, a faster SAT tester to solve many kinds of SAT problem is obtained.

Parallel Approximation Algorithms by Positive Linear Programming

Several sequential approximation algorithms for combinatorial optimization problems are based on the following paradigm: solve a linear or semidefinite programming relaxation, then use randomized rounding to convert fractional solutions of the relaxation into integer solutions for the original combinatorial problem. We demonstrate that such a paradigm can also yield parallel approximation algorithms by showing how to convert certain linear programming relaxations into essentially equivalent positive linear programming [LN] relaxations that can be near-optimally solved in NC. Building on this technique, and finding some new linear programming relaxations, we develop improved parallel approximation algorithms for Max Sat, Max Directed Cut, and Maxk CSP. The Max Sat algorithm essentially matches the best approximation obtainable with sequential algorithms and has a fast sequential version. The Maxk CSP algorithm improves even over previous sequential algorithms. We also show a connection between probabilistic proof checking and a restricted version of Maxk CSP. This implies that our approximation algorithm for Maxk CSP can be used to prove inclusion in P for certain PCP classes.

A hopeful CNF-SAT algorithm— Its high efficiency, industrial application and limitation

From the SAT physical model, a physical hypothesis named PHHY is proposed. By PHHY, it is proved that there is a universally efficient algorithm for solving SAT problem. Then, by square packing problem, the authors show that there are interesting industrial NP-complete problems which can be solved through SAT algorithms, but each way of solving like this will be much worse than that of a certain direct solving.

Meta-resolution: An algorithmic formalisation

We study the Tseitin formulae (Tseitin, 1970) on which resolution is known to be inefficient. We first give a new proof of the satisfiability condition and precise the number of solutions if any and otherwise the minimal contradictory subformula. Then, defining a new data structure, we introduce the meta-argument of that proof in the Davis-Putnam and Loveland procedure (Loveland, 1978) and obtain a new algorithm for SAT especially efficient on these formulae.