On testing convexity and submodularity

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 = {v_i,j}^i=n1,j=n2 _i,j=0 is called a Monge matrix if for every 0 ≤ i < r ≤ n₁ and 0 ≤ j < s ≤ n₂, we have v_i,j + v_r,s ≤ v^i,s + v_r,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)/ ε), where ε 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)/ε).