On testing convexity and submodularity

Convex and submodular 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,j=0^i=n1,j=n2 is called a Monge matrix if for every 0 ≤ i < i′ ≤ n₁ and 0 ≤ j < j′ ≤ n₂ we have v_i,j + v_i′,j′ ≤ v_i′,j′ + v_i′,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 testing algorithms for the above properties. A testing algorithm for a predetermined property P is given query access to an unknown function f and a distance parameter ε. The algorithm should accept f with high probability if it has the property P and reject it with high probability if more than an ε-fraction of the function values should be modified so that f obtains the property. Our algorithm for testing whether a 1-dimensional function f : [n] → ℝ is convex (concave) has query complexity and running time of O ((log n)/ε). Our algorithm for testing whether an n₁ × n₂ matrix V is a Monge (inverse Monge) matrix has query complexity and running time of O ((log n₁ · log n₂)/ε).