On the Equivalence among Problems of Bounded Width

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 $O^{\ast }(2^{\alpha \mathrm{t}\mathrm{w}})$ time,
• 3-SAT can be solved in $O^{\ast }(2^{\alpha \mathrm{t}\mathrm{w}})$ time,
• Max 2-SAT can be solved in $O^{\ast }(2^{\alpha \mathrm{t}\mathrm{w}})$ time,
• Independent Set can be solved in $O^{\ast }(2^{\alpha \mathrm{t}\mathrm{w}})$ time, and
• Independent Set can be solved in $O^{\ast }(2^{\alpha \mathrm{c}\mathrm{w}})$ 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 $O^{\ast }(c^k)$ time, then any problem in EPNL can be solved in $O^{\ast }(c^k)$ time.