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The Complexity of Homomorphism Reconstructibility

Authors Jan Böker , Louis Härtel , Nina Runde , Tim Seppelt , Christoph Standke

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Author Details

Jan Böker
  • RWTH Aachen University, Germany
Louis Härtel
  • RWTH Aachen University, Germany
Nina Runde
  • RWTH Aachen University, Germany
Tim Seppelt
  • RWTH Aachen University, Germany
Christoph Standke
  • RWTH Aachen University, Germany

Funding

  • Böker, Jan: European Union (ERC, SymSim, 101054974).
  • Runde, Nina: German Research Foundation (DFG) - 453349072.
  • Seppelt, Tim: European Union (ERC, SymSim, 101054974), German Research Foundation (DFG) - GRK 2236/2 (UnRAVeL).
  • Standke, Christoph: German Research Foundation (DFG) - GR 1492/16-1, GRK 2236/2 (UnRAVeL).

Acknowledgements

We would like to thank Martin Grohe, Athena Riazsadri, and Jan Tönshoff for fruitful discussions.

Cite AsGet BibTex

Jan Böker, Louis Härtel, Nina Runde, Tim Seppelt, and Christoph Standke. The Complexity of Homomorphism Reconstructibility. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 19:1-19:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.STACS.2024.19

Abstract

Representing graphs by their homomorphism counts has led to the beautiful theory of homomorphism indistinguishability in recent years. Moreover, homomorphism counts have promising applications in database theory and machine learning, where one would like to answer queries or classify graphs solely based on the representation of a graph G as a finite vector of homomorphism counts from some fixed finite set of graphs to G. We study the computational complexity of the arguably most fundamental computational problem associated to these representations, the homomorphism reconstructability problem: given a finite sequence of graphs and a corresponding vector of natural numbers, decide whether there exists a graph G that realises the given vector as the homomorphism counts from the given graphs. We show that this problem yields a natural example of an NP^#𝖯-hard problem, which still can be NP-hard when restricted to a fixed number of input graphs of bounded treewidth and a fixed input vector of natural numbers, or alternatively, when restricted to a finite input set of graphs. We further show that, when restricted to a finite input set of graphs and given an upper bound on the order of the graph G as additional input, the problem cannot be NP-hard unless 𝖯 = NP. For this regime, we obtain partial positive results. We also investigate the problem’s parameterised complexity and provide fpt-algorithms for the case that a single graph is given and that multiple graphs of the same order with subgraph instead of homomorphism counts are given.

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
  • counting complexity
  • parameterised complexity

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