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DOI: 10.4230/LIPIcs.STACS.2026.45

The Complexity of Homomorphism Reconstruction Revisited

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

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

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.

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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)

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@InProceedings{gervens_et_al:LIPIcs.STACS.2026.45,
  author =	{Gervens, Timo and Grohe, Martin and H\"{a}rtel, Louis and da Silva Fonseca, Philipp},
  title =	{{The Complexity of Homomorphism Reconstruction Revisited}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{45:1--45:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.45},
  URN =		{urn:nbn:de:0030-drops-255342},
  doi =		{10.4230/LIPIcs.STACS.2026.45},
  annote =	{Keywords: graph homomorphism, nexp-complete, counting complexity}
}

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The Complexity of Homomorphism Reconstruction Revisited

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