Lower Bounds for Dynamic Programming on Planar Graphs of Bounded Cutwidth

TL;DR

This paper establishes SETH-based lower bounds for solving Independent Set and Dominating Set on graphs with bounded cutwidth, including planar graphs, showing planarity does not improve computational complexity for these problems.

Contribution

It extends lower bounds from treewidth to cutwidth and planar graphs, demonstrating inherent computational hardness for these problems under SETH.

Findings

Independent Set cannot be solved in $O^*((2-initevarepsilon)^{cutw})$ time.

Dominating Set cannot be solved in $O^*((3-initevarepsilon)^{cutw})$ time.

Planarity does not reduce the complexity of solving these problems on graphs with bounded cutwidth.

Abstract

Many combinatorial problems can be solved in time $O^*(c^{tw})$ on graphs of treewidth $tw$, for a problem-specific constant $c$. In several cases, matching upper and lower bounds on $c$ are known based on the Strong Exponential Time Hypothesis (SETH). In this paper we investigate the complexity of solving problems on graphs of bounded cutwidth, a graph parameter that takes larger values than treewidth. We strengthen earlier treewidth-based lower bounds to show that, assuming SETH, Independent Set cannot be solved in $O^*((2-\varepsilon)^{cutw})$ time, and Dominating Set cannot be solved in $O^*((3-\varepsilon)^{cutw})$ time. By designing a new crossover gadget, we extend these lower bounds even to planar graphs of bounded cutwidth or treewidth. Hence planarity does not help when solving Independent Set or Dominating Set on graphs of bounded width. This sharply contrasts the fact that in many settings, planarity allows problems to be solved much more efficiently.