On solving the MAX-SAT using sum of squares

We consider semidefinite programming (SDP) approaches for solving the maximum satisfiability problem (MAX-SAT) and the weighted partial MAX-SAT. It is widely known that SDP is well-suited to approximate the (MAX-)2-SAT. Our work shows the potential of SDP also for other satisfiability problems, by being competitive with some of the best solvers in the yearly MAX-SAT competition. Our solver combines sum of squares (SOS) based SDP bounds and an efficient parser within a branch & bound scheme. On the theoretical side, we propose a family of semidefinite feasibility problems, and show that a member of this family provides the rank two guarantee. We also provide a parametric family of semidefinite relaxations for the MAX-SAT, and derive several properties of monomial bases used in the SOS approach. We connect two well-known SDP approaches for the (MAX)-SAT, in an elegant way. Moreover, we relate our SOS-SDP relaxations for the partial MAX-SAT to the known SAT relaxations.Comment: 26 pages, 5 figures, 8 tables, 2 appendix page

CNF Satisfiability in a Subspace and Related Problems

We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of polynomials over ?? each of which is a product of affine forms. We focus on the case of k-CNF formulas (the k-Sub-Sat problem). Clearly, k-Sub-Sat is no easier than k-SAT, and might be harder. Indeed, via simple reductions we show that 2-Sub-Sat is NP-hard, and W[1]-hard when parameterized by the co-dimension of the subspace. We also prove that the optimization version Max-2-Sub-Sat is NP-hard to approximate better than the trivial 3/4 ratio even on satisfiable instances. On the algorithmic front, we investigate fast exponential algorithms which give non-trivial savings over brute-force algorithms. We give a simple branching algorithm with running time (1.5)^r for 2-Sub-Sat, where r is the subspace dimension, as well as an O^*(1.4312)? time algorithm where n is the number of variables. Turning to k-Sub-Sat for k ? 3, while known algorithms for solving a system of degree k polynomial equations already imply a solution with running time ? 2^{r(1-1/2k)}, we explore a more combinatorial approach. Based on an analysis of critical variables (a key notion underlying the randomized k-SAT algorithm of Paturi, Pudlak, and Zane), we give an algorithm with running time ? {n choose {?t}} 2^{n-n/k} where n is the number of variables and t is the co-dimension of the subspace. This improves upon the running time of the polynomial equations approach for small co-dimension. Our combinatorial approach also achieves polynomial space in contrast to the algebraic approach that uses exponential space. We also give a PPZ-style algorithm for k-Sub-Sat with running time ? 2^{n-n/2k}. This algorithm is in fact oblivious to the structure of the subspace, and extends when the subspace-membership constraint is replaced by any constraint for which partial satisfying assignments can be efficiently completed to a full satisfying assignment. Finally, for systems of O(n) polynomial equations in n variables over ??, we give a fast exponential algorithm when each polynomial has bounded degree irreducible factors (but can otherwise have large degree) using a degree reduction trick

Fuzzy Maximum Satisfiability

In this paper, we extend the Maximum Satisfiability (MaxSAT) problem to {\L}ukasiewicz logic. The MaxSAT problem for a set of formulae {\Phi} is the problem of finding an assignment to the variables in {\Phi} that satisfies the maximum number of formulae. Three possible solutions (encodings) are proposed to the new problem: (1) Disjunctive Linear Relations (DLRs), (2) Mixed Integer Linear Programming (MILP) and (3) Weighted Constraint Satisfaction Problem (WCSP). Like its Boolean counterpart, the extended fuzzy MaxSAT will have numerous applications in optimization problems that involve vagueness.Comment: 10 page