A branch and cut algorithm for MAX-SAT and weighted MAX-SAT

DIMACS Series in Discrete Mathematicsand Theoretical Computer ScienceVolume00, 19xxA Branch and Cut Algorithm for MAX-SAT and WeightedMAX-SATSteveJoy, John Mitchell, and Brian BorchersAbstract.We describ e a branch and cut algorithm for b oth MAX-SAT andweighted MAX-SAT. This algorithm uses the GSAT pro cedure as a primalheuristic. At eachnode we solve a linear programming (LP) relaxation of theproblem. Twostyles of separating cuts are added: resolution cuts and o ddcycle inequalities.We compare our algorithm to an extension of the Davis Putnam Love-land (EDPL) algorithm. Our algorithm is more e ective than EDPL on someproblems, notably MAX-2-SAT. EDPL is more e ective on some other classesof problems.1. Intro ductionThe satis ability problem (SAT) is a problem in prop ositional logic. A logicformula consists of the conjunction of clauses. Each clause consists of a disjunctionof literals. Each literal is a variable or its negation. The SAT problem seeks to ndan assignment to the variables which satis es the logic formula, or an indicationthat no such assignment exists. The satis ability problem is NP-complete [5].There are a numb er of exact algorithms for the satis ability problem. Theseinclude Davis-Putnam-Loveland [4,23], resolution [26], and integer programmingapproaches [1,16, 18, 20,21]. A numb er of heuristics that use randomizationalso exist; the rst randomized lo cal search algorithm for satis abilitywas due toGu [9, 10, 11, 12]. Other algorithms include the GSAT heuristic [29, 30] and theGRASP heuristic [25]. For surveys of algorithms for SAT problems see [13,14].In this pap er weinvestigate the related MAX-SAT problem. Given a collectionof clauses, we seek a variable assignment that maximizes the numb er of satis edclauses. The weighted MAX-SAT problem assigns a wt to each clause, and seeksan assignment that maximizes the sum of the weights of the satis ed clauses. Bothof these problems are NP-hard. It is p ossible to approximate MAX-SAT within afactor of 1.325 in p olynomial time [6].Most SAT heuristics have b een extended to MAX-SAT. Several heuristics forMAX-SAT are summarized in Hansen and Jaumard [15]. The GSAT heuristic hasalso b een extended to weighted MAX-SAT[22].1991Mathematics Subject Classi cation.03B05, 49M35, 65K05, 90C10.Research supp orted by ONR Grant N00014{94{1{0391 to Rensselaer Polytechnic Institute.c0000 American Mathematical So ciety1052-1798/00 $1.00 + $.25 p er page1