Eth-tight algorithms for long path and cycle on unit disk graphs

Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Meirav Zehavi

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

3 Scopus citations

Abstract

We present an algorithm for the extensively studied Long Path and Long Cycle problems on unit disk graphs that runs in time 2O(√k)(n + m). Under the Exponential Time Hypothesis, Long Path and Long Cycle on unit disk graphs cannot be solved in time 2o(√k)(n + m)O(1) [de Berg et al., STOC 2018], hence our algorithm is optimal. Besides the 2O(√k)(n + m)O(1)-time algorithm for the (arguably) much simpler Vertex Cover problem by de Berg et al. [STOC 2018] (which easily follows from the existence of a 2k-vertex kernel for the problem), this is the only known ETH-optimal fixed-parameter tractable algorithm on UDGs. Previously, Long Path and Long Cycle on unit disk graphs were only known to be solvable in time 2O(√k log k)(n + m). This algorithm involved the introduction of a new type of a tree decomposition, entailing the design of a very tedious dynamic programming procedure. Our algorithm is substantially simpler: we completely avoid the use of this new type of tree decomposition. Instead, we use a marking procedure to reduce the problem to (a weighted version of) itself on a standard tree decomposition of width O(k).

Original languageEnglish
Title of host publication36th International Symposium on Computational Geometry, SoCG 2020
EditorsSergio Cabello, Danny Z. Chen
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771436
DOIs
StatePublished - 1 Jun 2020
Event36th International Symposium on Computational Geometry, SoCG 2020 - Zurich, Switzerland
Duration: 23 Jun 202026 Jun 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume164
ISSN (Print)1868-8969

Conference

Conference36th International Symposium on Computational Geometry, SoCG 2020
Country/TerritorySwitzerland
CityZurich
Period23/06/2026/06/20

Keywords

  • ETH
  • Long Cycle
  • Long Path
  • Optimality Program
  • Parameterized Complexity
  • Unit Disk Graphs

ASJC Scopus subject areas

  • Software

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