## Combination Can Be Hard: Approximability of the Unique Coverage Problem (2006)

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Venue: | In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms |

Citations: | 60 - 2 self |

### BibTeX

@INPROCEEDINGS{Demaine06combinationcan,

author = {Erik D. Demaine and Uriel Feige and Mohammadtaghi Hajiaghayi and Mohammad R. Salavatipour},

title = {Combination Can Be Hard: Approximability of the Unique Coverage Problem},

booktitle = {In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms},

year = {2006},

pages = {162--171}

}

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### Abstract

Abstract We prove semi-logarithmic inapproximability for a maximization problem called unique coverage:given a collection of sets, find a subcollection that maximizes the number of elements covered exactly once. Specifically, assuming that NP 6 ` BPTIME(2n " ) for an arbitrary "> 0, we prove O(1 / logoe n) inapproximability for some constant oe = oe("). We also prove O(1 / log1/3- " n) inapproximability, forany "> 0, assuming that refuting random instances of 3SAT is hard on average; and prove O(1 / log n)inapproximability under a plausible hypothesis concerning the hardness of another problem, balanced bipartite independent set. We establish an \Omega (1 / log n)-approximation algorithm, even for a moregeneral (budgeted) setting, and obtain an \Omega (1 / log B)-approximation algorithm when every set hasat most B elements. We also show that our inapproximability results extend to envy-free pricing, animportant problem in computational economics. We describe how the (budgeted) unique coverage problem, motivated by real-world applications, has close connections to other theoretical problemsincluding max cut, maximum coverage, and radio broadcasting. 1 Introduction In this paper we consider the approximability of the following natural maximization analog of set cover: Unique Coverage Problem. Given a universe U = {e1,..., en} of elements, and given a collection S = {S1,..., Sm} of subsets of U. Find a subcollection S0 ` S to maximize the number of elements that are uniquely covered, i.e., appear in exactly one set of S 0.