Decomposition of map graphs with applications

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

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

10 Scopus citations

Abstract

Bidimensionality is the most common technique to design subexponential-time parameterized algorithms on special classes of graphs, particularly planar graphs. The core engine behind it is a combinatorial lemma of Robertson, Seymour and Thomas that states that every planar graph either has a k × k-grid as a minor, or its treewidth is O(k). However, bidimensionality theory cannot be extended directly to several well-known classes of geometric graphs like unit disk or map graphs. This is mainly due to the presence of large cliques in these classes of graphs. Nevertheless, a relaxation of this lemma has been proven useful for unit disk graphs. Inspired by this, we prove a new decomposition lemma for map graphs, the intersection graphs of finitely many simply-connected and interior-disjoint regions of the Euclidean plane. Informally, our lemma states the following. For any map graph G, there exists a collection (U1,..., Ut) of cliques of G with the following property: G either contains a k × k-grid as a minor, or it admits a tree decomposition where every bag is the union of O(k) cliques in the above collection. The new lemma appears to be a handy tool in the design of subexponential parameterized algorithms on map_ graphs. We demonstrate its usability by designing algorithms on map graphs with running time 2O(k log k) · nO(1) for Connected Planar F-Deletion (that encompasses problems such as Feedback Vertex Set and Vertex Cover). Obtaining subexponential algorithms for Longest Cycle/Path and Cycle Packing is more challenging. We have to construct tree decompositions with more powerful properties and to prove sublinear bounds on the number of ways an optimum solution could “cross” bags in these decompositions. For Longest Cycle/Path, these are the first subexponential-time parameterized algorithm on map graphs. For Feedback Vertex Set and Cycle Packing, we improve upon known 2O(k0.75 log k) · nO(1)-time algorithms on map graphs.

Original languageEnglish
Title of host publication46th International Colloquium on Automata, Languages, and Programming, ICALP 2019
EditorsChristel Baier, Ioannis Chatzigiannakis, Paola Flocchini, Stefano Leonardi
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages60:1-60:15
ISBN (Electronic)9783959771092
DOIs
StatePublished - 1 Jul 2019
Event46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 - Patras, Greece
Duration: 9 Jul 201912 Jul 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume132
ISSN (Print)1868-8969

Conference

Conference46th International Colloquium on Automata, Languages, and Programming, ICALP 2019
Country/TerritoryGreece
CityPatras
Period9/07/1912/07/19

Keywords

  • Cycle Packing
  • FPT
  • Feedback Vertex Set
  • Longest Cycle
  • Map Graphs

ASJC Scopus subject areas

  • Software

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