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Decomposition of Map Graphs with Applications

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

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Author Details

Fedor V. Fomin
  • University of Bergen, Norway
Daniel Lokshtanov
  • University of California, Santa Barbara, USA
Fahad Panolan
  • University of Bergen, Norway
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel

Acknowledgements

This work is supported by the European Research Council (ERC) via grant LOPPRE, reference 819416, the Norwegian Research Council via project MULTIVAL, and Israel Science Foundation individual research grant no. 1176/18.

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Fedor V. Fomin, Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, and Meirav Zehavi. Decomposition of Map Graphs with Applications. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 60:1-60:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.60

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 sqrt{k} x sqrt{k}-grid as a minor, or its treewidth is O(sqrt{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 (U_1,...,U_t) of cliques of G with the following property: G either contains a sqrt{k} x sqrt{k}-grid as a minor, or it admits a tree decomposition where every bag is the union of O(sqrt{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 2^{O({sqrt{k}log{k}})} * n^{O(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 2^{O({k^{0.75}log{k}})} * n^{O(1)}-time algorithms on map graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Computational geometry
Keywords
  • Longest Cycle
  • Cycle Packing
  • Feedback Vertex Set
  • Map Graphs
  • FPT

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