Sums of squares, satisfiability and maximum satisfiability

Recently the Mathematical Programming community showed a renewed interest in Hilbert’s Positivstellensatz. The reason for this is that global optimization of polynomials in IR[x1,..., xn] is N P-hard, while the question whether a polynomial can be written as a sum of squares has tractable aspects. This is due to the fact that Semidefinite Programming can be used to decide in polynomial time (up to a prescribed precision) whether a polynomial can be rewritten as a sum of squares of linear combinations of monomials coming from a specified set. We investigate this approach in the context of Satisfiability. We examine the potential of the theory for providing tests for unsatisfiability and providing MAXSAT upper bounds. We compare the SOS (Sums Of Squares) approach with existing upper bound and rounding techniques for the MAX-2-SAT case of Goemans and Williamson [10] and Feige and Goemans [8] and the MAX-3-SAT case of Karloff and Zwick [12], which are based on Semidefinite Programming as well. We prove that for each of these algorithms there is a SOS-based counterpart which provides upper bounds at least as tight, but observably tighter in particular cases. Also, we propose a new randomized rounding technique based on the optimal solution of the SOS semidefinite program which we experimentally compare with the appropriate existing rounding techniques.