QCSP on Reflexive Tournaments

Authors Benoît Larose, Petar Marković, Barnaby Martin, Daniël Paulusma, Siani Smith, Stanislav Živný

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Author Details

Benoît Larose
  • LACIM, University of Québec, Montréal, Canada
Petar Marković
  • Department of Mathematics and Informatics, University of Novi Sad, Serbia
Barnaby Martin
  • Department of Computer Science, Durham University, UK
Daniël Paulusma
  • Department of Computer Science, Durham University, UK
Siani Smith
  • Department of Computer Science, Durham University, UK
Stanislav Živný
  • Department of Computer Science, University of Oxford, UK


We are grateful to several referees for careful reading of the paper and good advice.

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Benoît Larose, Petar Marković, Barnaby Martin, Daniël Paulusma, Siani Smith, and Stanislav Živný. QCSP on Reflexive Tournaments. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 58:1-58:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well-known that reflexive tournaments can be split into a sequence of strongly connected components H₁,…,H_n so that there exists an edge from every vertex of H_i to every vertex of H_j if and only if i < j. We prove that if H has both its initial and final strongly connected component (possibly equal) of size 1, then QCSP(H) is in NL and otherwise QCSP(H) is NP-hard.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • computational complexity
  • algorithmic graph theory
  • quantified constraints
  • universal algebra
  • constraint satisfaction


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