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 ℝ[x1,...,xn] is $\mathcal{NP}$-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 [8] and Feige and Goemans [6] and the MAX-3-SAT case of Karloff and Zwick [9], which are based on Semidefinite Programming as well. We show that the combination of the existing rounding techniques with the SOS based upper bound techniques yields polynomial time algorithms with a performance guarantee at least as good as the existing ones, but observably better in particular cases.