Abstract
A coverage function f over a ground set [m] is associated with a universe U of weighted elements and m sets A 1,…,A m ⊆ U, and for any T ⊆ [m], f(T) is defined as the total weight of the elements in the union ∪ j ∈ T A j . 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.
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Chakrabarty, D., Huang, Z. (2012). Testing Coverage Functions. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds) Automata, Languages, and Programming. ICALP 2012. Lecture Notes in Computer Science, vol 7391. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31594-7_15
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DOI: https://doi.org/10.1007/978-3-642-31594-7_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31593-0
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