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 $V = \left\{ {v_{i,j} } \right\}_{i,j = 0}^{i = n_1 ,j = n_2 }$ is called a Monge matrix if for every 0 ≤i < r ≤n ₁ and 0 ≤j < s ≤n ₂, 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((logn₁-logrn₂/∈), 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 ((logn)/∈).

Supported by the Israel Science Foundation (grant number 32/00-1).