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Restricted CSPs and F-Free Digraph Algorithmics

Authors Santiago Guzmán-Pro , Barnaby Martin

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LIPIcs.ICALP.2025.158.pdf
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ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Logic
Keywords
  • Digraph homomorphisms
  • constraint satisfaction problems
  • subgraph-free algorithmics

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Abstract

In recent years, much attention has been placed on the complexity of graph homomorphism problems when the input is restricted to ℙ_k-free and ℙ_k-subgraph-free graphs. We consider the directed version of this research line, by addressing the question is it true that digraph homomorphism problems CSP(H) have a P versus NP-complete dichotomy when the input is restricted to ℙ→_k-free (resp. ℙ→_k-subgraph-free) digraphs? Our main contribution in this direction shows that if CSP(H) is NP-complete, then there is a positive integer N such that CSP(H) remains NP-hard even for ℙ→_N-subgraph-free digraphs. Moreover, CSP(H) becomes polynomial-time solvable for ℙ→_{N-1}-subgraph-free acyclic digraphs. We then verify the questions above for digraphs on three vertices and a family of smooth tournaments. We prove these results by establishing a connection between F-(subgraph)-free algorithmics and constraint satisfaction theory. On the way, we introduce restricted CSPs, i.e., problems of the form CSP(H) restricted to yes-instances of CSP(H') - these were called restricted homomorphism problems by Hell and Nešetřil. Another main result of this paper presents a P versus NP-complete dichotomy for these problems. Moreover, this complexity dichotomy is accompanied by an algebraic dichotomy in the spirit of the finite domain CSP dichotomy.

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Santiago Guzmán-Pro and Barnaby Martin. Restricted CSPs and F-Free Digraph Algorithmics. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 158:1-158:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.158

Author Details

Santiago Guzmán-Pro
  • Institut für Algebra, TU Dresden, Germany
Barnaby Martin
  • Department of Computer Science, Durham University, UK

Funding

  • Guzmán-Pro, Santiago: Funded by the European Research Council (Project POCOCOP, ERC Synergy Grant 101071674). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
  • Martin, Barnaby: Funded by EPSRC Grant EP/X03190X/1.

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