Testing Real-Valued Modularity and Submodularity

We study the question of testing whether a function f : {0, 1} → R is modular/submodular or ε-far from it (with respect to Hamming distance). We provide two results: First, it is possible to test using O( n ε logn ) queries whether f is modular (equivalently affine, or linear with a constant term). For constant ε, this improves upon a simple tester that uses O(n) queries. Second, we prove that testing submodularity requires Ω( n 2 logn ) queries, thus separating the two problems. This improves on a linear lower bound due to Seshadhri and Vondrák. ∗Institute for Advanced Study, Princeton, NJ, USA. email: pooyahat@math.ias.edu Research supported by the National Science Foundation grant No. CCF-1412958 †Dept. of Mathematics, Stanford University, Stanford, CA 94305. email: jvondrak@stanford.edu