On Solving MAX-SAT Using Sum of Squares

Abstract

We consider semidefinite programming (SDP) approaches for solving the maximum satisfiability (MAX-SAT) problem and weighted partial MAX-SAT. It is widely known that SDP is well-suited to approximate (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-and-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 MAX-SAT and derive several properties of monomial bases used in the SOS approach. We connect two well-known SDP approaches for (MAX)-SAT in an elegant way. Moreover, we relate our SOS-SDP relaxations for partial MAX-SAT to the known SAT relaxations. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.0036 .