Time-Approximation Trade-offs for Inapproximable Problems

Authors Édouard Bonnet, Michael Lampis, Vangelis Th. Paschos



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Édouard Bonnet
Michael Lampis
Vangelis Th. Paschos

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Édouard Bonnet, Michael Lampis, and Vangelis Th. Paschos. Time-Approximation Trade-offs for Inapproximable Problems. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.STACS.2016.22

Abstract

In this paper we focus on problems which do not admit a constant-factor approximation in polynomial time and explore how quickly their approximability improves as the allowed running time is gradually increased from polynomial to (sub-)exponential. We tackle a number of problems: For MIN INDEPENDENT DOMINATING SET, MAX INDUCED PATH, FOREST and TREE, for any r(n), a simple, known scheme gives an approximation ratio of r in time roughly r^{n/r}. We show that, for most values of r, if this running time could be significantly improved the ETH would fail. For MAX MINIMAL VERTEX COVER we give a non-trivial sqrt{r}-approximation in time 2^{n/{r}}. We match this with a similarly tight result. We also give a log(r)-approximation for MIN ATSP in time 2^{n/r} and an r-approximation for MAX GRUNDY COLORING in time r^{n/r}. Furthermore, we show that MIN SET COVER exhibits a curious behavior in this super-polynomial setting: for any delta>0 it admits an m^delta-approximation, where m is the number of sets, in just quasi-polynomial time. We observe that if such ratios could be achieved in polynomial time, the ETH or the Projection Games Conjecture would fail.
Keywords
  • Algorithm
  • Complexity
  • Polynomial and Subexponential Approximation
  • Reduction
  • Inapproximability

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