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 V | 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.
One central object of this theory is the concept of M ♮ -concave functions, introduced by Murota and Shioura REF .
M-Convex Function on Generalized Polymatroid
Math. Oper. Res.
Math. Oper. Res.
Mathematics Computer Science