Testing submodularity and other properties of valuation functions

Abstract: We show that for any constant $\epsilon > 0$ and $p \ge 1$, it is possible to distinguish functions $f : {0,1}^n \to [0,1]$ that are submodular from those that are $\epsilon$-far from every submodular function in $\ell_p$ distance with a constant number of queries. More generally, we extend the testing-by-implicit-learning framework of Diakonikolas et al. (2007) to show that every property of real-valued functions that is well-approximated in $\ell_2$ distance by a class of $k$-juntas for some $k = O(1)$ can be tested in the $\ell_p$-testing model with a constant number of queries. This result, combined with a recent junta theorem of Feldman and Vondrak (2016), yields the constant-query testability of submodularity. It also yields constant-query testing algorithms for a variety of other natural properties of valuation functions, including fractionally additive (XOS) functions, OXS functions, unit demand functions, coverage functions, and self-bounding functions.