On the Equivalence among Problems of Bounded Width

TL;DR

This paper establishes the equivalence of various bounded-width problems in parameterized complexity, introduces a new complexity class EPNL, and shows certain problems are EPNL-complete, highlighting their computational interrelations.

Contribution

It introduces decomposition-based reductions for problem equivalence, defines the EPNL class, and proves EPNL-completeness for key problems parameterized by path-width.

Findings

Bounded-width problems are equivalent under certain conditions.

EPNL includes Set Cover and Directed Hamiltonicity.

Certain problems are EPNL-complete, linking their complexity.

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 $\alpha > 0$: * SAT can be solved in $O^*(2^{\alpha \mathrm{tw}})$ time, * 3-SAT can be solved in $O^*(2^{\alpha \mathrm{tw}})$ time, * Max 2-SAT can be solved in $O^*(2^{\alpha \mathrm{tw}})$ time, * Independent Set can be solved in $O^*(2^{\alpha \mathrm{tw}})$ time, and * Independent Set can be solved in $O^*(2^{\alpha \mathrm{cw}})$ 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 Directed Hamiltonicity, and show that SAT, 3-SAT, Max 2-SAT, and Independent Set parameterized by path-width are EPNL-complete. This implies that if one of these EPNL-complete problems can be solved in $O^*(c^k)$ time, then any problem in EPNL can be solved in $O^*(c^k)$ time.