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”, SIAM Journal on Computing, volume 38, number 4, September 2008, pages 1464–1483.

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 ⊈ BPTIME(2^nε) for an arbitrary ε > 0, we prove O(1/log^σ n) inapproximability for some constant σ = σ(ε). 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 an Ω(1/log n)-approximation algorithm, even for a more general (budgeted) setting, and obtain 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.

Comments:

This paper is also available from SIAM.

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