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Improved (In-)Approximability Bounds for d-Scattered Set (1910.05589v1)

Published 12 Oct 2019 in cs.CC and cs.DS

Abstract: 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 $\Delta{\lfloor d/2\rfloor-\epsilon}$ on the approximation ratio of any polynomial-time algorithm for graphs of maximum degree $\Delta$ and an improved upper bound of $O(\Delta{\lfloor d/2\rfloor})$ on the approximation ratio of any greedy scheme for this problem. - A polynomial-time $2\sqrt{n}$-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 $\rho$-approximation algorithm of (roughly) $2{\frac{n{1-\epsilon}}{\rho d}}$ for even $d$ and $2{\frac{n{1-\epsilon}}{\rho(d+\rho)}}$ for odd $d$ (under the randomized ETH), complemented by $\rho$-approximation algorithms of running times that (almost) match these bounds.

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