A Constructive Proof for the Induction of M-convex Functions through Networks

Murota (1995) introduced an M-convex function as a quantitative generalization of the set of integral vectors in an integral base polyhedron as well as an extension of valuated matroid over base polyhedron. Just as a base polyhedron can be transformed through a network, an M-convex function can be induced through a network. This paper gives a constructive proof for the induction of an M-convex function. The proof is based on the correctness of a simple algorithm, which nds an exchangeable element. We also analyze a behavior of induced functions when they take the value 1: