Testing Submodularity and Other Properties of Valuation Functions

We show that for any constant ǫ > 0 and p ≥ 1, it is possible to distinguish functions f : {0, 1} → [0, 1] that are submodular from those that are ǫ-far from every submodular function in lp 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 l2 distance by a class of k-juntas for some k = O(1) can be tested in the lp-testing model with a constant number of queries. This result, combined with a recent junta theorem of Feldman and Vondrák (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.