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Time-Approximation Trade-offs for Inapproximable Problems

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Date
2016
Dewey
Recherche opérationnelle
Sujet
Approximation; Complexity; Polynomial and Subexponential Approximation; Reduction; Inapproximability
DOI
http://dx.doi.org/10.4230/LIPIcs.STACS.2016.22
Conference name
33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)
Conference date
02-2016
Conference city
Orléans
Conference country
France
Book title
33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)
Author
Ollinger, Nicolas; Vollmer, Heribert
Publisher
Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik
Publisher city
Wadern
Year
02-2016
Pages number
798
ISBN
978-3-95977-001-9
Book URL
10.4230/LIPIcs.STACS.2016.0
URI
https://basepub.dauphine.fr/handle/123456789/16336
Collections
  • LAMSADE : Publications
Metadata
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Author
Bonnet, Édouard
220549 Institute for Computer Science and Control [Budapest] [SZTAKI]
Lampis, Michael
989 Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Paschos, Vangelis
989 Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Type
Communication / Conférence
Item number of pages
22:1-22:14
Abstract (EN)
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.

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