Counting Homomorphisms in Plain Exponential Time

Authors Andrei A. Bulatov, Amineh Dadsetan

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Andrei A. Bulatov
  • School of Computing Science, Simon Fraser University, Burnaby, Canada
Amineh Dadsetan
  • School of Computing Science, Simon Fraser University, Burnaby, Canada

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Andrei A. Bulatov and Amineh Dadsetan. Counting Homomorphisms in Plain Exponential Time. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


In the counting Graph Homomorphism problem (#GraphHom) the question is: Given graphs G,H, find the number of homomorphisms from G to H. This problem is generally #P-complete, moreover, Cygan et al. proved that unless the Exponential Time Hypothesis fails there is no algorithm that solves this problem in time O(|V(H)|^o(|V(G)|)). This, however, does not rule out the possibility that faster algorithms exist for restricted problems of this kind. Wahlström proved that #GraphHom can be solved in plain exponential time, that is, in time O((2k+1)^(|V(G)|+|V(H)|) poly(|V(H)|,|V(G)|)) provided H has clique width k. We generalize this result to a larger class of graphs, and also identify several other graph classes that admit a plain exponential algorithm for #GraphHom.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • graph homomorphisms
  • plain exponential time
  • clique width


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