Combination Can Be Hard: Approximability of the Unique Coverage Problem

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(sigma/(epsilon)) n) inapproximability assuming that NP not subset of BPTIME(2(n epsilon)) for some epsilon > 0. We also prove O(1/log(1/3-epsilon) n) inapproximability, for any epsilon > 0, assuming that refuting random instances of 3SAT is hard on average; and prove 0(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 Omega(1/log n)-approximation algorithm as well as an Omega(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.