Hard Satisfiability Problems Research Articles

Clustering of solutions in hard satisfiability problems

We study numerically the solution space structure of random 3-SAT problemsclose to the SAT/UNSAT transition. This is done by considering chains ofsatisfiability problems, where clauses are added sequentially to a problem instance.Using the overlap measure of similarity between different solutions found onthe same problem instance, we examine geometrical changes as a function ofα. In each chain, the overlap distribution is first smooth, but then develops a tiered structure,indicating that the solutions are found in well separated clusters. On chains of not too largeinstances, all remaining solutions are eventually observed to be found in only one small clusterbefore vanishing. This condensation transition point is estimated by finite size scaling to beαc = 4.26 with an apparent critical exponent of about1.7. The average overlap value is also observed to increase withα up to the transition, indicating a reduction in solutions space size, in accordancewith theoretical predictions. The solutions are generated by a local heuristic,ASAT, and compared to those found by the Survey Propagation algorithm up toαc.

Behavior of heuristics on large and hard satisfiability problems

We study the behavior of a heuristic for solving random satisfiability problems by stochastic local search near the satisfiability threshold. The heuristic for average satisfiability (ASAT), is similar to the Focused Metropolis Search heuristic, and shares the property of being focused, i.e., only variables in unsatisfied clauses are updated in each step. It is significantly simpler than the benchmark WALKSAT heuristic. We show that ASAT solves instances as large as N=10(6) in linear time, on average, up to a ratio of 4.21 clauses per variable in random three-satisfiability. For K higher than 3, ASAT appears to solve instances of K -satisfiability up to the Montanari-Ricci-Tersenghi-Parisi full replica symmetry breaking (FSRB) threshold denoted alpha(s)(K) in linear time.

Solving Non-Boolean Satisfiability Problems with Stochastic Local Search: A Comparison of Encodings

Much excitement has been generated by the success of stochastic local search procedures at finding solutions to large, very hard satisfiability problems. Many of the problems on which these procedures have been effective are non-Boolean in that they are most naturally formulated in terms of variables with domain sizes greater than two. Approaches to solving non-Boolean satisfiability problems fall into two categories. In the direct approach, the problem is tackled by an algorithm for non-Boolean problems. In the transformation approach, the non-Boolean problem is reformulated as an equivalent Boolean problem and then a Boolean solver is used. This paper compares four methods for solving non-Boolean problems: one direct and three transformational. The comparison first examines the search spaces confronted by the four methods, and then tests their ability to solve random formulas, the round-robin sports scheduling problem, and the quasigroup completion problem. The experiments show that the relative performance of the methods depends on the domain size of the problem and that the direct method scales better as domain size increases. Along the route to performing these comparisons we make three other contributions. First, we generalize Walksat, a highly successful stochastic local search procedure for Boolean satisfiability problems, to work on problems with domains of any finite size. Second, we introduce a new method for transforming non-Boolean problems to Boolean problems and improve on an existing transformation. Third, we identify sufficient conditions for omitting at-least-one and at-most-one clauses from a transformed formula. Fourth, for use in our experiments we propose a model for generating random formulas that vary in domain size but are similar in other respects.

Optimization by quantum annealing: Lessons from hard satisfiability problems

The path integral Monte Carlo simulated quantum annealing algorithm is applied to the optimization of a large hard instance of the random satisfiability problem (N = 10,000). The dynamical behavior of the quantum and the classical annealing are compared, showing important qualitative differences in the way of exploring the complex energy landscape of the combinatorial optimization problem. At variance with the results obtained for the Ising spin glass and for the traveling salesman problem, in the present case the linear-schedule quantum annealing performance is definitely worse than classical annealing. Nevertheless, a quantum cooling protocol based on field-cycling and able to outperform standard classical simulated annealing over short time scales is introduced.

A two-phase algorithm for solving a class of hard satisfiability problems

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.

Parallel Satisfiability Test with Synchronous Simulated Annealing on Distributed-Memory Multiprocessor

Solving the hard Satisfiability Problem is time consuming even for modest-sized problem instances. Solving the Random L-SAT Problem is especially difficult due to the ratio of clauses to variables. This report presents a practical approach to solving the Random L-SAT Problem on a large-scale distributed-memory multiprocessor. In particular, we use a parallel synchronous simulated annealing procedure, called Generalized Speculative Computation, which guarantees the same decision sequence as sequential simulated annealing. We have selected problem instances varying in size from 100-variable/425-clause to 5000-variable/21,250-clause. Experimental results on the AP1000 distributed-memory multiprocessor indicate that Generalized Speculative Computation of synchronous simulated annealing can satisfy 99.9% of the clauses while giving almost a 70-fold speedup on 500 processors.

Generating hard satisfiability problems

We report results from large-scale experiments in satisfiability testing. As has been observed by others, testing the satisfiability of random formulas often appears surprisingly easy. Here we show that by using the right distribution of instances, and appropriate parameter values, it is possible to generate random formulas that are hard, that is, for which satisfiability testing is quite difficult. Our results provide a benchmark for the evaluation of satisfiability testing procedures.