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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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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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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.

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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

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.

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