The Exponential-Time Complexity of Counting (Quantum) Graph Homomorphisms

  • Hubie Chen
  • Radu CurticapeanEmail author
  • Holger Dell
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11789)


Many graph parameters can be expressed as homomorphism counts to fixed target graphs; this includes the number of independent sets and the number of k-colorings for any fixed k. Dyer and Greenhill (RSA 2000) gave a sweeping complexity dichotomy for such problems, classifying which target graphs render the problem polynomial-time solvable or \(\#\mathrm {P}\)-hard. In this paper, we give a new and shorter proof of this theorem, with previously unknown tight lower bounds under the exponential-time hypothesis. We similarly strengthen complexity dichotomies by Focke, Goldberg, and Živný (SODA 2018) for counting surjective homomorphisms to fixed graphs. Both results crucially rely on our main contribution, a complexity dichotomy for evaluating linear combinations of homomorphism numbers to fixed graphs. In the terminology of Lovász (Colloquium Publications 2012), this amounts to counting homomorphisms to quantum graphs.


Graph homomorphisms Exponential-time hypothesis Counting complexity Complexity dichotomy Surjective homomorphisms 


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Authors and Affiliations

  1. 1.Department of Computer Science and Information SystemsBirkbeck University of LondonLondonUK
  2. 2.Basic Algorithms Research Copenhagen (BARC)CopenhagenDenmark
  3. 3.IT University of CopenhagenCopenhagenDenmark

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