Combination can be hard: approximability of the unique coverage problem

We prove semi-logarithmic inapproximability for a maximization problem called <i>unique coverage</i>: given a collection of sets, find a subcollection that maximizes the number of elements covered exactly once. Specifically, we prove <i>O</i>(1/ log^σ(ε)<i>n</i>) inapproximability assuming that NP ⊈ BPTIME(2^<i>n</i>ε) for some ε > 0. We also prove <i>O</i>(1/log^1/3-ε <i>n</i>) inapproximability, for any ε > 0, assuming that refuting random instances of 3SAT is hard on average; and prove <i>O</i>(1/log <i>n</i>) 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 <i>n</i>)-approximation algorithm as well as an Ω(1/log <i>B</i>)-approximation algorithm when every set has at most <i>B</i> 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 broad-casting.