Compression of M${}^\natural$-convex Functions -- Flag Matroids and Valuated Permutohedra

TL;DR

This paper introduces a new compression operation on M${}^ atural$-convex functions, transforming them into M-convex functions and revealing structural insights into valuated matroids and permutohedra.

Contribution

It defines a convolution-based compression method for M${}^ atural$-convex functions and analyzes its structural implications for valuated matroids and permutohedra.

Findings

The compression transforms M${}^ atural$-convex functions into M-convex functions.

Valuated generalized matroids decompose into flag-matroid strips via the induced permutohedron.

Structural analysis of flag-matroid strips and permutohedra using discrete convex analysis.

Abstract

Murota (1998) and Murota and Shioura (1999) introduced concepts of M-convex function and M${}^\natural$-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${}^\natural$-convex function that transforms the given M${}^\natural$-convex function to an M-convex function, which we call a compression of an M${}^\natural$-convex function. For the class of valuated generalized matroids, which are special M${}^\natural$-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.