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The Complexity of Finding Fair Independent Sets in Cycles

Author Ishay Haviv

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Ishay Haviv
  • School of Computer Science, The Academic College of Tel Aviv-Yaffo, Tel Aviv, Israel


We are grateful to Aris Filos-Ratsikas and Alexander Golovnev for helpful discussions.

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Ishay Haviv. The Complexity of Finding Fair Independent Sets in Cycles. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 4:1-4:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


Let G be a cycle graph and let V₁,…,V_m be a partition of its vertex set into m sets. An independent set S of G is said to fairly represent the partition if |S ∩ V_i| ≥ 1/2⋅|V_i| - 1 for all i ∈ [m]. It is known that for every cycle and every partition of its vertex set, there exists an independent set that fairly represents the partition (Aharoni et al., A Journey through Discrete Math., 2017). We prove that the problem of finding such an independent set is PPA-complete. As an application, we show that the problem of finding a monochromatic edge in a Schrijver graph, given a succinct representation of a coloring that uses fewer colors than its chromatic number, is PPA-complete as well. The work is motivated by the computational aspects of the "cycle plus triangles" problem and of its extensions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Graph theory
  • Fair independent sets in cycles
  • the complexity class {PPA}
  • Schrijver graphs


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