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

  • Ioannis KatsikarelisEmail author
  • Michael Lampis
  • Vangelis Th. Paschos
Conference paper
  • 82 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11926)

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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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Ioannis Katsikarelis
    • 1
    Email author
  • Michael Lampis
    • 1
  • Vangelis Th. Paschos
    • 1
  1. 1.Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243 LAMSADEParisFrance

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