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

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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)


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
  • Hadwiger Number
  • Exponential-Time Hypothesis
  • Exact Algorithms
  • Edge Contraction Problems


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