License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2016.22
URN: urn:nbn:de:0030-drops-57236
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5723/
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Bonnet, Édouard ; Lampis, Michael ; Paschos, Vangelis Th.

Time-Approximation Trade-offs for Inapproximable Problems

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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.

BibTeX - Entry

@InProceedings{bonnet_et_al:LIPIcs:2016:5723,
  author =	{{\'E}douard Bonnet and Michael Lampis and Vangelis Th. Paschos},
  title =	{{Time-Approximation Trade-offs for Inapproximable Problems}},
  booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
  pages =	{22:1--22:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-001-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{47},
  editor =	{Nicolas Ollinger and Heribert Vollmer},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5723},
  URN =		{urn:nbn:de:0030-drops-57236},
  doi =		{10.4230/LIPIcs.STACS.2016.22},
  annote =	{Keywords: Algorithm, Complexity, Polynomial and Subexponential Approximation, Reduction, Inapproximability}
}

Keywords: Algorithm, Complexity, Polynomial and Subexponential Approximation, Reduction, Inapproximability
Collection: 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)
Issue Date: 2016
Date of publication: 16.02.2016

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