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ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
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Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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A polynomial-time approximation scheme for the geometric unique coverage problem on unit squares
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
A path-decomposition theorem with applications to pricing and covering on trees
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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Proceedings of the 23rd international conference on World wide web
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We prove semilogarithmic 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 $\mathrm{NP}\not\subseteq\operatorname{BPTIME}(2^{n^\varepsilon})$ for an arbitrary $\varepsilon0$, we prove $O(1/\log^{\sigma}n)$ inapproximability for some constant $\sigma=\sigma(\varepsilon)$. We also prove $O(1/\log^{1/3-\varepsilon}n)$ inapproximability for any $\varepsilon0$, assuming that refuting random instances of 3SAT is hard on average; and we prove $O(1/\log n)$ inapproximability under a plausible hypothesis concerning the hardness of another problem, balanced bipartite independent set. We establish an $\Omega(1/\log n)$-approximation algorithm, even for a more general (budgeted) setting, and obtain 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.