Testing Coverage Functions

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 2Θ~(m)">2Θ~(m) queries to distinguish it from the class of coverage functions.