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

TL;DR

This paper investigates the approximability of the d-Scattered Set problem, providing new bounds and algorithms that improve understanding of its computational complexity and approximation limits.

Contribution

It offers improved bounds on approximation ratios for d-Scattered Set, extending known results for Independent Set, and introduces algorithms matching these bounds under ETH assumptions.

Findings

Lower bound of elta^{\u2212psilon} on approximation ratio for graphs with max degree elta.

A polynomial-time 2 approximation for bipartite graphs with even d.

Complexity bounds for pproximation algorithms under randomized ETH.

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.