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Going Beyond Twin-Width? CSPs with Unbounded Domain and Few Variables

Authors Peter Jonsson , Victor Lagerkvist , Jorke M. de Vlas , Magnus Wahlström

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LIPIcs.ICALP.2026.120.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Constraint and logic programming
Keywords
  • Constraint satisfaction problems
  • parameterized complexity
  • twin-width
  • universal algebra

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Abstract

We study connections between parameterized complexity, universal algebra, and structural graph parameters. Our starting point is the constraint satisfaction problem over instances with few variables but unbounded domain size (udCSP). Surprisingly, many upper and lower bounds in parameterized complexity can be expressed as solving such udCSPs. Prominent examples include the FPT algorithms for Boolean MinCSP [Eun Jung Kim et al., 2025], Directed Multicut with three cut requests [Meike Hatzel et al., 2023], and the canonical W[1]-hardness construction Paired Min Cut [Dániel Marx and Igor Razgon, 2009]. We represent constraints over unbounded domains by a set of unary maps ℳ into a finite base language Γ, situating udCSP(Γ, ℳ) in the algebraic terra incognita between finite and infinite domains. We present a novel algebraic theory that explains the parameterized complexity of problems such as Paired Min Cut, 𝓁-Chain Sat, and Coupled Min Cut, and unifies disparate FPT algorithms through the lens of twin-width. In particular, we simplify key steps in existing algorithms, e.g., for Boolean MinCSP, via a clean reduction to udCSP. We specifically concentrate on udCSP(Γ,ℳ) restricted to monotone maps Mo, where we identify the crucial connector polymorphism: its presence implies FPT for binary relations (via dynamic programming based on twin-width), while its absence entails W[1]-hardness. Extending this to higher-arity relations is related to the notoriously difficult task of finding a generalisation of twin-width to non-binary structures. As a step in this direction, inspired by our algebraic framework, we introduce a new structural parameter, projected grid-rank, and show that it coincides with the connector property, and agrees with twin-width for binary structures. More strongly, we show that for structures of bounded arity and bounded projected grid-rank, all binary projections have bounded twin-width. This width measure may thus be of independent interest for any problem currently hinging on generalizations of twin-width.

Cite As Get BibTex

Peter Jonsson, Victor Lagerkvist, Jorke M. de Vlas, and Magnus Wahlström. Going Beyond Twin-Width? CSPs with Unbounded Domain and Few Variables. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 120:1-120:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.120

Author Details

Peter Jonsson
  • Linköping University, Sweden
Victor Lagerkvist
  • Linköping University, Sweden
Jorke M. de Vlas
  • Linköping University, Sweden
Magnus Wahlström
  • Royal Holloway, University of London, Egham, UK

Funding

  • Jonsson, Peter: Partially supported by the Swedish Research Council (VR) under grant 2021-04371.
  • Lagerkvist, Victor: Partially supported by the Swedish Research Council (VR) under grant 2025-04487.
  • de Vlas, Jorke M.: Partially supported by the Swedish Research Council (VR) under grant 2021-04371.

Acknowledgements

We thank the anonymous reviewers for several helpful comments.

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