Paper by Erik D. Demaine

Reference:

Erik D. Demaine, Uriel Feige, MohammadTaghi Hajiaghayi, and Mohammad R. Salavatipour, “Combination Can Be Hard: Approximability of the Unique Coverage Problem”, in Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), Miami, Florida, January 22–24, 2006, pages 162–171.

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, we prove O(1/log^σ(ε) n) inapproximability assuming that NP ⊈ BPTIME(2^nε) for some ε > 0. We also prove O(1/log^1/3−ε n) inapproximability, for any ε > 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 matching upper bounds up to exponents, even for a more general (budgeted) setting, giving an Ω(1/log n)-approximation algorithm as well as an Ω(1/log B)-approximation algorithm when every set has at most B elements. We also show that our inapproximability results extend to envy-free pricing, an important problem in computational economics. We describe how the (budgeted) unique coverage problem, motivated by real-world applications, has close connections to other theoretical problems including max cut, maximum coverage, and radio broadcasting.

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