Reducing hard SAT instances to polynomial ones

2002

Abstract Conventional algorithms for Boolean satisfiability (SAT) work within the framework of resolution as a proof system. However it has been known since the 1980s that every resolution proof of the pigeonhole principle (PHPm n), suitably encoded as a CNF instance, includes exponentially many steps [4]. Therefore SAT solvers based upon the DLL procedure [1] or the DP procedure [2] must take exponential time.

Software Engineering [SEN], 1998

The DIMACS suite of satisfiability (SAT) benchmarks contains a set of instances that are very hard for existing algorithms. These instances arise from learning the parity function on 32 bits. In this paper we develop a two-phase algorithm that is capable of solving these instances. In the first phase, a polynomially solvable subproblem is identified and solved. Using the solution to this problem, we can considerably restrict the size of the search space in the second phase of the algorithm, which is an extension of the well-known Davis-Putnam-Logemann-Loveland algorithm. We conclude with reporting on our computational results on the parity instances.

2002

The availability of decision procedures for combinations of boolean and linear mathematical propositions opens the ability to solve problems arising from real-world domains such as verification of timed systems and planning with resources. In this paper we present a general and efficient approach to the problem, based on two main ingredients. The first is a DPLL-based SAT procedure, for dealing efficiently with the propositional component of the problem. The second is a tight integration, within the DPLL architecture, of a set of mathematical deciders for theories of increasing expressive power. A preliminary experimental evaluation shows the potential of the approach.

The Journal of Logic Programming, 1989

Let p be a set of n atomic propositions, which can be either true or false, and denote by T a proposition which is always true, and by F a proposition which is always false. Let ~tf be a set of m clauses of the form PI VPz V . . . VP, -P,+IAP. .,2A . . . APq , GIORGIO GALLO AND GIAMPAOLO URBANI to be NP-complete [2], although it is not one of the hardest problems in that class ; in fact, under certain hypotheses it has been proven to have a linear average case complexity .

arXiv (Cornell University), 2019

When creating benchmarks for SAT solvers, we need SAT instances that are easy to build but hard to solve. A recent development in the search for such methods has led to the Balanced SAT algorithm, which can create k-SAT instances with m clauses of high difficulty, for arbitrary k and m. In this paper we introduce the No-Triangle SAT algorithm, a SAT instance generator based on the cluster coefficient graph statistic. We empirically compare the two algorithms by fixing the arity and the number of variables, but varying the number of clauses. The hardest instances that we find are produced by No-Triangle SAT. Furthermore, difficult instances from No-Triangle SAT have a different number of clauses than difficult instances from Balanced SAT, potentially allowing a combination of the two methods to find hard SAT instances for a larger array of parameters.

Information Processing Letters, 1988

2007

Proposing a fibre view on propositional clause sets, we investigate satisfiability testing for several CNF subclasses. Specifically, we show how to decide SAT in polynomial time for formulas where each pair of different clauses intersect either in all or in one variable

In this paper I describe the progress, preliminary results and future work directions of a project of implementing a many-valued SAT solver based on a generalization of algorithms used in modern Boolean SAT solvers. Mimicking Boolean SAT solvers minimizes the algorithm-design and implementation challenges related to such a task, since many ideas can be easily adapted to the many-valued setting. Experimental results show that even on the early stages of the development a many-valued solver can perform better on some problems than modern Boolean SAT solvers.

1995

We introduce a greedy local search procedure called GSAT for solving propositional satis ability problems. Our experiments show that this procedure can be used to solve hard, randomly generated problems that are an order of magnitude larger than those that can be handled by more traditional approaches such as the Davis-Putnam procedure or resolution. We also show that GSAT can solve structured satis ability problems quickly. In particular, we solve encodings of graph coloring problems, N-queens, and Boolean induction. General application strategies and limitations of the approach are also discussed.

Design Automation …, 2001

Boolean Satisfiability is probably the most studied of combinatorial optimization/search problems. Significant effort has been devoted to trying to provide practical solutions to this problem for problem instances encountered in a range of applications in Electronic Design Automation (EDA), as well as in Artificial Intelligence (AI). This study has culminated in the development of several SAT packages, both proprietary and in the public domain (e.g. GRASP, SATO) which find significant use in both research and industry. Most existing complete solvers are variants of the Davis-Putnam (DP) search algorithm. In this paper we describe the development of a new complete solver, Chaff, which achieves significant performance gains through careful engineering of all aspects of the search-especially a particularly efficient implementation of Boolean constraint propagation (BCP) and a novel low overhead decision strategy. Chaff has been able to obtain one to two orders of magnitude performance improvement on difficult SAT benchmarks in comparison with other solvers (DP or otherwise), including GRASP and SATO.