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URN: urn:nbn:de:0030-drops-64114
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### Finding Large Set Covers Faster via the Representation Method

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### Abstract

The worst-case fastest known algorithm for the Set Cover problem on universes with n elements still essentially is the simple O^*(2^n)-time dynamic programming algorithm, and no non-trivial consequences of an O^*(1.01^n)-time algorithm are known. Motivated by this chasm, we study the following natural question: Which instances of Set Cover can we solve faster than the simple dynamic programming algorithm? Specifically, we give a Monte Carlo algorithm that determines the existence of a set cover of size sigma*n in O^*(2^{(1-Omega(sigma^4))n}) time. Our approach is also applicable to Set Cover instances with exponentially many sets: By reducing the task of finding the chromatic number chi(G) of a given n-vertex graph G to Set Cover in the natural way, we show there is an O^*(2^{(1-Omega(sigma^4))n})-time randomized algorithm that given integer s = sigma*n, outputs NO if chi(G) > s and YES with constant probability if \chi(G) <= s - 1.

On a high level, our results are inspired by the "representation method" of Howgrave-Graham and Joux~[EUROCRYPT'10] and obtained by only evaluating a randomly sampled subset of the table entries of a dynamic programming algorithm.

### BibTeX - Entry

@InProceedings{nederlof:LIPIcs:2016:6411,
author =	{Jesper Nederlof},
title =	{{Finding Large Set Covers Faster via the Representation Method}},
booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
pages =	{69:1--69:15},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-015-6},
ISSN =	{1868-8969},
year =	{2016},
volume =	{57},
editor =	{Piotr Sankowski and Christos Zaroliagis},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},