Improved (In-)Approximability Bounds for d-Scattered Set
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In the d-Scattered Set problem we are asked to select at least k vertices of a given graph, so that the distance between any pair is at least d. We study the problem's (in-)approximability and offer improvements and extensions of known results for Independent Set, of which the problem is a generalization. Specifically, we show: - A lower bound of Δ^ d/2-ϵ on the approximation ratio of any polynomial-time algorithm for graphs of maximum degree Δ and an improved upper bound of O(Δ^ d/2) on the approximation ratio of any greedy scheme for this problem. - A polynomial-time 2√(n)-approximation for bipartite graphs and even values of d, that matches the known lower bound by considering the only remaining case. - A lower bound on the complexity of any ρ-approximation algorithm of (roughly) 2^n^1-ϵ/ρ d for even d and 2^n^1-ϵ/ρ(d+ρ) for odd d (under the randomized ETH), complemented by ρ-approximation algorithms of running times that (almost) match these bounds.
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