Exact MAX 2-SAT: Easier and faster

Abstract

Prior algorithms known for exactly solving MAX 2-SAT improve upon the trivial upper bound only for very sparse instances. We present new algorithms for exactly solving (in fact, counting) weighted MAX 2-SAT instances. One of them has a good performance if the underlying constraint graph has a small separator decomposition, another has a slightly improved worst case performance. For a 2-SAT instance F with n variables, the worst case running time is Õ(21-1/(d̃(F)_1))n), where d̃(F) is the average degree in the constraint graph defined by F. We use strict α-gadgets introduced by Trevisan, Sorkin, Sudan, and Williamson to get the same upper bounds for problems like MAX 3-SAT and MAX CUT. We also introduce a notion of strict (α, β)-gadget to provide a framework that allows composition of gadgets. This framework allows us to obtain the same upper bounds for MAX k-SAT and MAX k-LIN-2. © Springer-Verlag Berlin Heidelberg 2007.