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Isolation Schemes for Problems on Decomposable Graphs

Authors Jesper Nederlof , Michał Pilipczuk , Céline M. F. Swennenhuis , Karol Węgrzycki

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Author Details

Jesper Nederlof
  • Utrecht University, The Netherlands
Michał Pilipczuk
  • University of Warsaw, Poland
Céline M. F. Swennenhuis
  • Eindhoven University of Technology, The Netherlands
Karol Węgrzycki
  • Saarland University, Saarbrücken, Germany
  • Max Planck Institute for Informatics, Saarbrücken, Germany

Funding

  • Nederlof, Jesper: Supported by the project CRACKNP (grant agreement No 853234) that has received funding from the European Research Council (ERC).
  • Pilipczuk, Michał: Supported by the project TOTAL (grant agreement No 677651) that has received funding from the European Research Council (ERC).
  • Swennenhuis, Céline M. F.: Supported by the Netherlands Organization for Scientific Research under project no. 613.009.031b.
  • Węgrzycki, Karol: Supported by the project TIPEA (grant agreement No 850979) that has received funding from the European Research Council (ERC).

Cite AsGet BibTex

Jesper Nederlof, Michał Pilipczuk, Céline M. F. Swennenhuis, and Karol Węgrzycki. Isolation Schemes for Problems on Decomposable Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 50:1-50:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.STACS.2022.50

Abstract

The Isolation Lemma of Mulmuley, Vazirani and Vazirani [Combinatorica'87] provides a self-reduction scheme that allows one to assume that a given instance of a problem has a unique solution, provided a solution exists at all. Since its introduction, much effort has been dedicated towards derandomization of the Isolation Lemma for specific classes of problems. So far, the focus was mainly on problems solvable in polynomial time. In this paper, we study a setting that is more typical for NP-complete problems, and obtain partial derandomizations in the form of significantly decreasing the number of required random bits. In particular, motivated by the advances in parameterized algorithms, we focus on problems on decomposable graphs. For example, for the problem of detecting a Hamiltonian cycle, we build upon the rank-based approach from [Bodlaender et al., Inf. Comput.'15] and design isolation schemes that use - 𝒪(tlog n + log²{n}) random bits on graphs of treewidth at most t; - 𝒪(√n) random bits on planar or H-minor free graphs; and - 𝒪(n)-random bits on general graphs. In all these schemes, the weights are bounded exponentially in the number of random bits used. As a corollary, for every fixed H we obtain an algorithm for detecting a Hamiltonian cycle in an H-minor-free graph that runs in deterministic time 2^{𝒪(√n)} and uses polynomial space; this is the first algorithm to achieve such complexity guarantees. For problems of more local nature, such as finding an independent set of maximum size, we obtain isolation schemes on graphs of treedepth at most d that use 𝒪(d) random bits and assign polynomially-bounded weights. We also complement our findings with several unconditional and conditional lower bounds, which show that many of the results cannot be significantly improved.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • Isolation Lemma
  • Derandomization
  • Hamiltonian Cycle
  • Exact Algorithms

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