Probabilistic Combinatorial Optimization

The (bounded) generalized maximum satisfiability problem covers a broad range of NP-complete problems, e.g. it is a generalization of INDEPENDENT SET, LINEAR INEQUALITY, HITTING SET, SET PACKING, MINIMUM COVER, etc. The complexity of finding approximations for problems in this class is analyzed. The results have several interpretations, including the following: - A general class of existence proofs is made efficiently constructive. - A class of randomized algorithms is made deterministic and efficient. - A new class of combinatorial approximation algorithms is introduced, which is based on “background” optimization. Instead of maximizing among all assignments we maximize among expected values for parametrized random solutions. It turns out that this “background” optimization is in two precise senses best possible if P≠NP. The “background optimization” performed is equivalent to finding the maximum of a polynomial in a bounded region.