Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
Introduction to Linear Optimization
Introduction to Linear Optimization
From convex optimization to randomized mechanisms: toward optimal combinatorial auctions
Proceedings of the forty-third annual ACM symposium on Theory of computing
Limitations of Randomized Mechanisms for Combinatorial Auctions
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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A coverage functionf over a ground set [m] is associated with a universe U of weighted elements and m sets A1,…,Am⊆U, and for any T⊆[m], f(T) is defined as the total weight of the elements in the union ∪j∈TAj. Coverage functions are an important special case of submodular functions, and arise in many applications, for instance as a class of utility functions of agents in combinatorial auctions. Set functions such as coverage functions often lack succinct representations, and in algorithmic applications, an access to a value oracle is assumed. In this paper, we ask whether one can test if a given oracle is that of a coverage function or not. We demonstrate an algorithm which makes O(m|U|) queries to an oracle of a coverage function and completely reconstructs it. This gives a polytime tester for succinct coverage functions for which |U| is polynomially bounded in m. In contrast, we demonstrate a set function which is "far" from coverage, but requires $2^{\tilde{\Theta}(m)}$ queries to distinguish it from the class of coverage functions.