On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime Order

Authors J. A. Gregor Lagodzinski , Andreas Göbel , Katrin Casel , Tobias Friedrich

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J. A. Gregor Lagodzinski
  • Hasso Plattner Institute, University of Potsdam, Germany
Andreas Göbel
  • Hasso Plattner Institute, University of Potsdam, Germany
Katrin Casel
  • Hasso Plattner Institute, University of Potsdam, Germany
Tobias Friedrich
  • Hasso Plattner Institute, University of Potsdam, Germany


The authors would like to thank Holger Dell for bringing [Hubie Chen et al., 2019] to their attention. Our gratitude also goes to Jacob Focke and Marc Roth for their valuable insights on partially surjective homomorphisms and for pointing out a mistake in a previous version.

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J. A. Gregor Lagodzinski, Andreas Göbel, Katrin Casel, and Tobias Friedrich. On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime Order. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 91:1-91:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We study the problem of counting the number of homomorphisms from an input graph G to a fixed (quantum) graph ̄{H} in any finite field of prime order ℤ_p. The subproblem with graph H was introduced by Faben and Jerrum [ToC'15] and its complexity is still uncharacterised despite active research, e.g. the very recent work of Focke, Goldberg, Roth, and Zivný [SODA'21]. Our contribution is threefold. First, we introduce the study of quantum graphs to the study of modular counting homomorphisms. We show that the complexity for a quantum graph ̄{H} collapses to the complexity criteria found at dimension 1: graphs. Second, in order to prove cases of intractability we establish a further reduction to the study of bipartite graphs. Lastly, we establish a dichotomy for all bipartite (K_{3,3}$1{e}, {domino})-free graphs by a thorough structural study incorporating both local and global arguments. This result subsumes all results on bipartite graphs known for all prime moduli and extends them significantly. Even for the subproblem with p = 2 this establishes new results.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Discrete mathematics
  • Algorithms
  • Theory
  • Quantum Graphs
  • Bipartite Graphs
  • Graph Homomorphisms
  • Modular Counting
  • Complexity Dichotomy


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