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.

No abstract

This paper is an in-depth analysis of the generalized ϑ-number of a graph. The generalized ϑ-number, ϑ k (G), serves as a bound for both the k-multichromatic number of a graph and the maximum k-colorable subgraph problem. We present various properties of ϑ k (G), such as that the series (ϑ k (G)) k is increasing and bounded above by the order of the graph G. We study ϑ k (G) when G is the graph strong, disjunction and Cartesian product of two graphs. We provide closed form expressions for the generalized ϑ-number on several classes of graphs including the Kneser graphs, cycle graphs, strongly regular graphs and orthogonality graphs. Our paper provides bounds on the product and sum of the k-multichromatic number of a graph and its complement graph, as well as lower bounds for the k-multichromatic number on several graph classes including the Hamming and Johnson graphs.