Operations on M-Convex Functions on Jump Systems

A jump system is a set of integer points with an exchange property, which is a generalization of a matroid, a delta-matroid, and a base polyhedron of an integral polymatroid (or a submodular system). Recently, the concept of M-convex functions on constant-parity jump systems is introduced by Murota as a class of discrete convex functions that admit a local criterion for global minimality. M-convex functions on constant-parity jump systems generalize valuated matroids, valuated delta-matroids, and M-convex functions on base polyhedra. This paper reveals that the class of M-convex functions on constant-parity jump systems is closed under a number of natural operations such as splitting, aggregation, convolution, composition, and transformation by networks. The present results generalize hitherto-known similar constructions for matroids, delta-matroids, valuated matroids, valuated delta-matroids, and M-convex functions on base polyhedra.