Recognizing Coverage Functions

A coverage function $f$ over a ground set $[m]$ is associated with a universe $U$ of weighted elements and $m$ sets $A_1,\ldots,A_m \subseteq U$, and for any $T\subseteq [m]$, $f(T)$ is defined as the total weight of the elements in the union $\cup_{j\in T} A_j$. 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. Naive representations of coverage functions have size exponential in $m$, and in algorithmic applications, an access to a value oracle is assumed. In this paper, we ask whether one can recognize 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 is polynomial time whenever $|U|$ is polynomially bounded implying the function has a succinct description. To complement the above result, we show a negative result. We prove that “no...