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Computation of Hadwiger Number and Related Contraction Problems: Tight Lower Bounds

Authors Fedor V. Fomin, Daniel Lokshtanov, Ivan Mihajlin, Saket Saurabh, Meirav Zehavi

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Author Details

Fedor V. Fomin
  • University of Bergen, Norway
Daniel Lokshtanov
  • University of California, Santa Barbara, CA, USA
Ivan Mihajlin
  • University of California, San Diego, CA, USA
Saket Saurabh
  • Department of Informatics, University of Bergen, Norway
  • The Institute of Mathematical Sciences, Chennai, India
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

  • Fomin, Fedor V.: Research Council of Norway via the project MULTIVAL.
  • Lokshtanov, Daniel: European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 715744), and United States - Israel Binational Science Foundation grant no. 2018302.
  • Saurabh, Saket: European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 819416), and Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.
  • Zehavi, Meirav: Israel Science Foundation grant no. 1176/18, and United States – Israel Binational Science Foundation grant no. 2018302.

Cite As Get BibTex

Fedor V. Fomin, Daniel Lokshtanov, Ivan Mihajlin, Saket Saurabh, and Meirav Zehavi. Computation of Hadwiger Number and Related Contraction Problems: Tight Lower Bounds. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 49:1-49:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.49

Abstract

We prove that the Hadwiger number of an n-vertex graph G (the maximum size of a clique minor in G) cannot be computed in time n^o(n), unless the Exponential Time Hypothesis (ETH) fails. This resolves a well-known open question in the area of exact exponential algorithms. The technique developed for resolving the Hadwiger number problem has a wider applicability. We use it to rule out the existence of n^o(n)-time algorithms (up to ETH) for a large class of computational problems concerning edge contractions in graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Hadwiger Number
  • Exponential-Time Hypothesis
  • Exact Algorithms
  • Edge Contraction Problems

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References

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