On the Equivalence among Problems of Bounded Width

Abstract

In this paper, we introduce a methodology, called decomposition-based 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 $O^*(2^{\alpha{\bf tw} })$ time,

• 3-SAT can be solved in $O^*(2^{\alpha{\bf tw} })$ time,

• Max 2-SAT can be solved in $O^*(2^{\alpha{\bf tw} })$ time,

• Independent Set can be solved in $O^*(2^{\alpha{\bf tw} })$ time, and

• Independent Set can be solved in $O^*(2^{\alpha{\bf cw} })$ time,

where ${\bf tw}$ and ${\bf cw}$ are the tree-width and clique-width of the instance, respectively. Then, we introduce a new parameterized complexity class ${\mbox{\textup{EPNL}}}$, which includes Set Cover and TSP, and show that SAT, 3-SAT, Max 2-SAT, and Independent Set parameterized by path-width are ${\mbox{\textup{EPNL}}}$-complete. This implies that if one of these ${\mbox{\textup{EPNL}}}$-complete problems can be solved in O ^*(c ^k) time, then any problem in ${\mbox{\textup{EPNL}}}$ can be solved in O ^*(c ^k) time.