On the equivalence among problems of bounded width

Abstract

In this paper, we introduce a methodology, called decompositionbased reductions, for showing the equivalence among various problems of bounded-width. First, we show that the following are equivalent for any α > 0: – SAT can be solved in O∗ (2αtw) time, – 3-SAT can be solved in O∗ (2αtw) time, – Max 2-SAT can be solved in O∗(2αtw) time, – Independent Set can be solved in O∗(2αtw) time, and – Independent Set can be solved in O∗ (2αcw) time, where tw and cw are the tree-width and clique-width of the instance, respectively. Then, we introduce a new parameterized complexity class EPNL, which includes Set Cover and TSP, and show that SAT, 3-SAT, Max 2-SAT, and Independent Set parameterized by path-width are EPNLcomplete. This implies that if one of these EPNL-complete problems can be solved in O∗ (ck) time, then any problem in EPNL can be solved in O∗(ck) time.