Chapter 24. Maximum Satisfiabiliy

Maximum satisfiability (MaxSAT) is an optimization version of SAT that is solved by finding an optimal truth assignment instead of just a satisfying one. In MaxSAT the objective function to be optimized is specified by a set of weighted soft clauses: the objective value of a truth assignment is the sum of the weights of the soft clauses it satisfies. In addition, the MaxSAT problem can have hard clauses that the truth assignment must satisfy. Many optimization problems can be naturally encoded into MaxSAT and this, along with significant performance improvements in MaxSAT solvers, has led to MaxSAT being used in a number of different application areas. This chapter provides a detailed overview of the approaches to MaxSAT solving that have in recent years been most successful in solving real-world optimization problems. Further recent developments in MaxSAT research are also overviewed, including encodings, applications, preprocessing, incomplete solving, algorithm portfolios, partitioning-based solving, and parallel solving.

FROM XSAT TO SAT BY EXHIBITING BOOLEAN FUNCTIONS

Given a Boolean formula in conjunctive normal form (CNF), the Exact Satisfiability problem (XSAT), a variant of the Satisfiability problem (SAT), consists in finding an assignment to the variables such that each clause contains exactly one satisfied literal. Best algorithms to solve this problem run in [Formula: see text] ([Formula: see text] for X3SAT). Another possibility is to transform each clause in a set of equivalent clauses for the Satisfiability problem and to use modern and powerful solvers (zChaff, Berkmin, MiniSat, RSat etc.) to find such truth assignment. In this paper we introduce three new encodings from XSAT instances to SAT instances that lead to a lot of structural information (equivalency gates and and gates) which is naturally hidden in the pairwise transformation. Some solvers (lsat,march_dl,eqsatz) can take into account this kinds of structural information to make simplifications as pretreatment and speed-up the resolution. Then we show the interest of dealing with the XSAT formalism by introducing an encoding of binary CSP and graph coloring problem into XSAT instances. Preliminary results on real-world binary CSP and graph coloring problem show the importance of exhibiting equivalencies for the XSAT problem.

Generating Hard Satisfiable Formulas by Hiding Solutions Deceptively

To test incomplete search algorithms for constraint satisfaction problems such as 3-SAT, we need a source of hard, but satisfiable, benchmark instances. A simple way to do this is to choose a random truth assignment A, and then choose clauses randomly from among those satisfied by A. However, this method tends to produce easy problems, since the majority of literals point toward the "hidden'' assignment A. Last year, Achlioptas, Jia and Moore proposed a problem generator that cancels this effect by hiding both A and its complement. While the resulting formulas appear to be just as hard for DPLL algorithms as random 3-SAT formulas with no hidden assignment, they can be solved by WalkSAT in only polynomial time. Here we propose a new method to cancel the attraction to A, by choosing a clause with t > 0 literals satisfied by A with probability proportional to q^t for some q < 1. By varying q, we can generate formulas whose variables have no bias, i.e., which are equally likely to be true or false; we can even cause the formula to "deceptively'' point away from A. We present theoretical and experimental results suggesting that these formulas are exponentially hard both for DPLL algorithms and for incomplete algorithms such as WalkSAT.

Hiding Satisfying Assignments: Two are Better than One

The evaluation of incomplete satisfiability solvers depends critically on the availability of hard satisfiable instances. A plausible source of such instances consists of random k-SAT formulas whose clauses are chosen uniformly from among all clauses satisfying some randomly chosen truth assignment A. Unfortunately, instances generated in this manner tend to be relatively easy and can be solved efficiently by practical heuristics. Roughly speaking, for a number of different algorithms, A acts as a stronger and stronger attractor as the formula's density increases. Motivated by recent results on the geometry of the space of satisfying truth assignments of random k-SAT and NAE-k-SAT formulas, we introduce a simple twist on this basic model, which appears to dramatically increase its hardness. Namely, in addition to forbidding the clauses violated by the hidden assignment A, we also forbid the clauses violated by its complement, so that both A and compliment of A are satisfying. It appears that under this "symmetrization" the effects of the two attractors largely cancel out, making it much harder for algorithms to find any truth assignment. We give theoretical and experimental evidence supporting this assertion.

New Algorithms for Exact Satisfiability

The Exact Satisfiability problem is to determine if a CNF-formula has a truth assignment satisfying exactly one literal in each clause; Exact 3-Satisfiability is the version in which each clause contains at most three literals. In this paper, we present algorithms for Exact Satisfiability and Exact 3-Satisfiability running in time O(2^{0.2325n}) and O(2^{0.1379n}), respectively. The previously best algorithms have running times O(2^{0.2441n}) for Exact Satisfiability (Monien, Speckenmeyer and Vornberger (1981)) and O(2^{0.1626n}) for Exact 3-Satisfiability (Kulikov and independently Porschen, Randerath and Speckenmeyer (2002)). We extend the case analyses of these papers and observe, that a formula not satisfying any of our cases has a small number of variables, for which we can try all possible truth assignments and for each such assignment solve the remaining part of the formula in polynomial time.