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DOI: 10.7155/jgaa.00542
Lower Bounds for Dynamic Programming on Planar Graphs of Bounded Cutwidth
Vol. 24, no. 3, pp. 461482, 2020. Regular paper.
Abstract Many combinatorial problems can be solved in time $\mathcal{O}^*(c^{\mathrm{tw}})$ on graphs of treewidth $\mathrm{tw}$, for a problemspecific 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 treewidthbased lower bounds to show that, assuming SETH, $\rm{I{\small NDEPENDENT}~S{\small ET}}$ cannot be solved in $O^*((2\varepsilon)^{\mathrm{ctw}})$ time, and $\rm{D{\small OMINATING}~S{\small ET}}$ cannot be solved in $O^*((3\varepsilon)^{\mathrm{ctw}})$ 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 $\rm{I{\small NDEPENDENT}~S{\small ET}}$ or $\rm{D{\small OMINATING}~S{\small ET}}$ on graphs of bounded width. This sharply contrasts the fact that in many settings, planarity allows problems to be solved much more efficiently.

Submitted: December 2018.
Reviewed: August 2020.
Revised: September 2020.
Accepted: September 2020.
Final: October 2020.
Published: October 2020.
Communicated by
Gerhard J. Woeginger
