On Testing Convexity and Submodularity

Abstract

Submodular and convex functions play an important role in many applications, and in particular in combinatorial optimization. Here we study two special cases: convexity in one dimension and submodularity in two dimensions. The latter type of functions are equivalent to the well known Monge matrices. A matrix V = {vi,j}i,j=0i=n1, j=n2 is called a Monge matrix if for every 0 ? i < r ? n1 and 0 ? j < s ? n2, we have vi,j + vr,s ? vi,s + vr,j. If inequality holds in the opposite direction then V is an inverse Monge matrix (supermodular function). Many problems, such as the traveling salesperson problem and various transportation problems, can be solved more efficiently if the input is a Monge matrix.In this work we present a testing algorithm for Monge and inverse Monge matrices, whose running time is O ((log n1 ? log n2)/?), where ? is the distance parameter for testing. In addition we have an algorithm that tests whether a function f : [n] ? R is convex (concave) with running time of O ((log n)/?).