NDLI: Nearly Optimal NP-Hardness of Unique Coverage

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Lee, Euiwoong Guruswami, Venkatesan
Copyright Year	2017
Abstract	The Unique Coverage problem, given a universe $V$ of elements and a collection $E$ of subsets of $V$, asks to find $S \subseteq V$ to maximize the number of $e \in E$ that intersects $S$ in exactly one element. When each $e \in E$ has cardinality at most $k$, it is also known as 1-in-$k$ Hitting Set and admits a simple $\Omega(\frac{1}{\log k})$-approximation algorithm. For constant $k$, we prove that 1-in-$k$ Hitting Set is NP-hard to approximate within a factor $O(\frac{1}{\log k})$. This improves the result of Guruswami and Zhou [Theory Comput., 8 (2012), pp. 239--267], who proved the same result assuming the Unique Games Conjecture. For Unique Coverage, we prove that it is hard to approximate within a factor $O(\frac{1}{\log^{1 - \epsilon} n})$ for any $\epsilon > 0$, unless NP admits quasi-polynomial time algorithms. This improves the results of Demaine et al. [SIAM J. Comput., 38 (2008), pp. 1464--1483], including their $\approx 1/\log^{1/3} n$ inapproximability factor, which was proven under the Random 3SAT Hypothesis. Our simple proof combines ideas from two classical inapproximability results for the Set Cover and Constraint Satisfaction Problems, made efficient by various derandomization methods based on bounded independence.
Sponsorship	Samsung. National Science Foundation
Starting Page	1018
Ending Page	1028
Page Count	11
File Format	PDF
ISSN	00975397
DOI	10.1137/16M1070682
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	3
Volume Number	46
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-06-13
Access Restriction	Subscribed
Subject Keyword	Computational difficulty of problems hardness of approximation Unique Coverage 1-in-$k$ Hitting Set
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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