On Finding Constrained Independent Sets in Cycles
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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.
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