Isolation Schemes for Problems on Decomposable Graphs

Jesper Nederlof, MichaŁ Pilipczuk, Céline M. F. Swennenhuis, Karol Wegrzycki

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

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.
Original languageEnglish
Title of host publication39th International Symposium on Theoretical Aspects of Computer Science, STACS 2022, March 15-18, 2022, Marseille, France (Virtual Conference)
EditorsPetra Berenbrink, Benjamin Monmege
PublisherSchloss Dagstuhl – Leibniz-Zentrum für Informatik GmbH
Pages50:1-50:20
ISBN (Print)978-3-95977-222-8
DOIs
Publication statusPublished - 2022

Publication series

NameLeibniz International Proceedings in Informatics
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
Volume219
ISSN (Print)1868-8969

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