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# On Finding Constrained Independent Sets in Cycles

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## Cite As

Ishay Haviv. On Finding Constrained Independent Sets in Cycles. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 73:1-73:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.73

## Abstract

A subset of [n] = {1,2,…,n} is called stable if it forms an independent set in the cycle on the vertex set [n]. In 1978, Schrijver proved via a topological argument that for all integers n and k with n ≥ 2k, the family of stable k-subsets of [n] cannot be covered by n-2k+1 intersecting families. We study two total search problems whose totality relies on this result. In the first problem, denoted by Schrijver(n,k,m), we are given an access to a coloring of the stable k-subsets of [n] with m = m(n,k) colors, where m ≤ n-2k+1, and the goal is to find a pair of disjoint subsets that are assigned the same color. While for m = n-2k+1 the problem is known to be PPA-complete, we prove that for m < d ⋅ ⌊n/(2k+d-2)⌋, with d being any fixed constant, the problem admits an efficient algorithm. For m = ⌊n/2⌋-2k+1, we prove that the problem is efficiently reducible to the Kneser problem. Motivated by the relation between the problems, we investigate the family of unstable k-subsets of [n], which might be of independent interest. In the second problem, called Unfair Independent Set in Cycle, we are given 𝓁 subsets V_1, …, V_𝓁 of [n], where 𝓁 ≤ n-2k+1 and |V_i| ≥ 2 for all i ∈ [𝓁], and the goal is to find a stable k-subset S of [n] satisfying the constraints |S ∩ V_i| ≤ |V_i|/2 for i ∈ [𝓁]. We prove that the problem is PPA-complete and that its restriction to instances with n = 3k is at least as hard as the Cycle plus Triangles problem, for which no efficient algorithm is known. On the contrary, we prove that there exists a constant c for which the restriction of the problem to instances with n ≥ c ⋅ k can be solved in polynomial time.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph theory
• Mathematics of computing → Combinatorial algorithms
• Theory of computation → Problems, reductions and completeness
##### Keywords
• Schrijver graph
• Kneser graph
• Stable sets
• PPA-completeness

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