M-Convex Function on Generalized Polymatroid

The concept of M-convex function, introduced recently by Murota, is a quantitative generalization of the set of integral points in an integral base polyhedron as well as an extension of valuated matroid of Dress–Wenzel (1990). In this paper, we extend this concept to functions on generalized polymatroids with a view to providing a unified framework for efficiently solvable nonlinear discrete optimization problems. The restriction of a function to {x ∈ Z | x(V ) = k} for k ∈ Z is called a layer. We prove the M-convexity of each layer, and reveal that the minimizers in consecutive layers are closely related. Exploiting these properties, we can solve the optimization on layers efficiently. A number of equivalent exchange axioms are given for M-convex function on generalized polymatroid.