**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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**Related papers**:- UniqueCoverage_SODA2006 (Combination Can Be Hard: Approximability of the Unique Coverage Problem)

See also other papers by Erik Demaine.

Last updated July 23, 2024 by Erik Demaine.