Compression of M♮-convex functions — Flag matroids and valuated permutohedra

Murota (1998) and Murota and Shioura (1999) introduced concepts of M-convex function and M♮-convex function as discrete convex functions, which are generalizations of valuated matroids due to Dress and Wenzel (1992). In the present paper we consider a new operation defined by a convolution of sections of an M♮-convex function that transforms the given M♮-convex function to an M-convex function, which we call a compression of an M♮-convex function. For the class of valuated generalized matroids, which are special M♮-convex functions, the compression induces a valuated permutohedron together with a decomposition of the valuated generalized matroid into flag-matroid strips, each corresponding to a maximal linearity domain of the induced valuated permutohedron. We examine the details of the structure of flag-matroid strips and the induced valuated permutohedron by means of discrete convex analysis of Murota.