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Time-Approximation Trade-offs for Inapproximable Problems

Authors Édouard Bonnet, Michael Lampis, Vangelis Th. Paschos

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Keywords
  • Algorithm
  • Complexity
  • Polynomial and Subexponential Approximation
  • Reduction
  • Inapproximability

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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.

Cite As Get BibTex

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

Author Details

Édouard Bonnet
Michael Lampis
Vangelis Th. Paschos

References

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