Abstract
Many combinatorial problems can be solved in time $\mathcal{O}^*(c^{\mathrm{tw}})$ on graphs of treewidth $\mathrm{tw}$, 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, $\rm{I{\small NDEPENDENT}~S{\small ET}}$ 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.
Highlights
Dynamic programming on graphs of bounded treewidth is a powerful tool in the algorithm designer’s toolbox, which has many applications and is captured by several metatheorems [7, 25]
In this work we have investigated whether Strong Exponential Time Hypothesis (SETH)-based lower bounds for solving problems on graphs of bounded treewidth apply for (1) planar graphs and (2) graphs of bounded cutwidth
We showed that the graph parameter cutwidth can be preserved when reducing to a planar instance using suitably restricted crossover gadgets
Summary
Dynamic programming on graphs of bounded treewidth is a powerful tool in the algorithm designer’s toolbox, which has many applications (cf. [5]) and is captured by several metatheorems [7, 25]. While the lower bound construction of Lokshtanov et al [23] works for the parameter cutwidth after a minor tweak, no crossover gadget for the Dominating Set problem was known. Our work resolves the question raised by Lokshtanov et al [23] and by Baste and Sau [1] whether the SETH-lower bounds for Independent Set and Dominating Set parameterized by treewidth apply for planar graphs. It leads to a proof of Theorem 1. Proofs for statements marked ( ) have been deferred to the full version [16]
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