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Limitations of Affine Integer Relaxations for Solving Constraint Satisfaction Problems

Authors Moritz Lichter , Benedikt Pago

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LIPIcs.ICALP.2025.166.pdf
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ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Complexity theory and logic
Keywords
  • constraint satisfaction
  • affine relaxation
  • promise CSPs
  • ℤ-affine k-consistency
  • cohomological k-consistency algorithm
  • Tseitin
  • graph isomorphism

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Abstract

We show that various recent algorithms for finite-domain constraint satisfaction problems (CSP), which are based on solving their affine integer relaxations, do not solve all tractable and not even all Maltsev CSPs. This rules them out as candidates for a universal polynomial-time CSP algorithm. The algorithms are ℤ-affine k-consistency, BLP+AIP, BA^{k}, and CLAP. We thereby answer a question by Brakensiek, Guruswami, Wrochna, and Živný [Joshua Brakensiek et al., 2020] whether a constant level of BA^{k}solves all tractable CSPs in the negative: Indeed, not even a sublinear level k suffices. We also refute a conjecture by Dalmau and Opršal [Víctor Dalmau and Jakub Opršal, 2024] (LICS 2024) that every CSP is either solved by ℤ-affine k-consistency or admits a Datalog reduction from 3-colorability. For the cohomological k-consistency algorithm, that is also based on affine relaxations, we show that it correctly solves our counterexample but fails on an NP-complete template.

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Moritz Lichter and Benedikt Pago. Limitations of Affine Integer Relaxations for Solving Constraint Satisfaction Problems. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 166:1-166:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.166

Author Details

Moritz Lichter
  • RWTH Aachen University, Germany
Benedikt Pago
  • University of Cambridge, UK

Funding

  • Lichter, Moritz: The author received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (SymSim: grant agreement No. 101054974).Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
  • Pago, Benedikt: Funded by UK Research and Innovation (UKRI) under the UK government’s Horizon Europe funding guarantee: grant number EP/X028259/1.

Acknowledgements

We thank a number of people for helpful discussions and valuable input at various stages of this work, especially also for acquainting us with the problem: We are grateful to Anuj Dawar, Martin Grohe, Andrei Krokhin, Adam Ó Conghaile, Jakub Opršal, Standa Živný, and Dmitriy Zhuk. We are especially indebted to Michael Kompatscher, who kindly provided us with the proof of Theorem 2 (3).

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