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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

Funding

Research supported in part by the Israel Science Foundation (grant No. 1218/20).

Acknowledgements

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

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ITCS.2021.4

Abstract

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
Keywords
  • Fair independent sets in cycles
  • the complexity class {PPA}
  • Schrijver graphs

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