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Lower Bounds for Dynamic Programming on Planar Graphs of Bounded Cutwidth

Authors Bas A. M. van Geffen, Bart M. P. Jansen , Arnoud A. W. M. de Kroon, Rolf Morel



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Author Details

Bas A. M. van Geffen
  • University of Oxford
Bart M. P. Jansen
  • Eindhoven University of Technology
Arnoud A. W. M. de Kroon
  • University of Oxford
Rolf Morel
  • University of Oxford

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Bas A. M. van Geffen, Bart M. P. Jansen, Arnoud A. W. M. de Kroon, and Rolf Morel. Lower Bounds for Dynamic Programming on Planar Graphs of Bounded Cutwidth. In 13th International Symposium on Parameterized and Exact Computation (IPEC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 115, pp. 3:1-3:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.IPEC.2018.3

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-epsilon)^{ctw}) time, and Dominating Set cannot be solved in O^*((3-epsilon)^{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 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • planarization
  • dominating set
  • cutwidth
  • lower bounds
  • strong exponential time hypothesis

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