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The Complexity of Homomorphism Reconstruction Revisited

Authors Timo Gervens , Martin Grohe , Louis Härtel , Philipp da Silva Fonseca

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Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
  • Mathematics of computing → Graph theory
Keywords
  • graph homomorphism
  • nexp-complete
  • counting complexity

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Abstract

We revisit the algorithmic problem of reconstructing a graph from homomorphism counts that has first been studied in (Böker et al., STACS 2024): given graphs F₁,…,F_k and counts m₁,…,m_k, decide if there is a graph G such that the number of homomorphisms from F_i to G is m_i, for all i. We prove that the problem is NEXP-hard if the counts m_i are specified in binary and Σ₂^p-complete if they are in unary.
Furthermore, as a positive result, we show that the unary version can be solved in polynomial time if the constraint graphs are stars of bounded size.

Cite As Get BibTex

Timo Gervens, Martin Grohe, Louis Härtel, and Philipp da Silva Fonseca. The Complexity of Homomorphism Reconstruction Revisited. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 45:1-45:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.45

Author Details

Timo Gervens
  • RWTH Aachen University, Germany
Martin Grohe
  • RWTH Aachen University, Germany
Louis Härtel
  • RWTH Aachen University, Germany
Philipp da Silva Fonseca
  • RWTH Aachen University, Germany

Funding

Timo Gervens, Martin Grohe, Louis Härtel: European Union (ERC, SymSim, 101054974).

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