On Finding Constrained Independent Sets in Cycles

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.