Testing submodularity and other properties of valuation functions

TL;DR

This paper demonstrates that submodularity and related properties of valuation functions can be tested with a constant number of queries using a novel framework, significantly advancing property testing in high-dimensional function spaces.

Contribution

It extends the testing-by-implicit-learning framework to real-valued functions, enabling constant-query testing of submodularity and other properties.

Findings

Constant-query testability of submodularity.

Testing algorithms for XOS, OXS, and coverage functions.

Extension of property testing framework to real-valued 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.