Vertex Structure of Master Corner Polyhedra

TL;DR

This paper studies the vertices of master corner polyhedra in integer linear programming, introducing combinatorial operations, support vertices, and characterizing their structure and properties.

Contribution

It introduces combinatorial operations transforming vertices, defines support vertices, and characterizes the polyhedra's structure and diameter, advancing the understanding of master corner polyhedra.

Findings

Support vertices form a basis for all vertices.

Polyhedra have diameter 2.

Support vertices are invariant under automorphisms.

Abstract

This paper focuses on vertices of the master corner polyhedra $P(G,g_0),$ the core of the group-theoretical approach to integer linear programming. We introduce two combinatorial operations that transform each vertex of $P(G,g_0)$ to adjacent ones. This implies that for any $P(G,g_0),$ there exists a subset of basic vertices, we call them support vertices, from which all others can be built. The class of support vertices is proved to be invariant under the automorphism group of $G,$ so this basis can be further reduced to a subset of pairwise non-equivalent support vertices. Among other results, we characterize irreducible points of the master corner polyhedra, establish relations between an integer point and the nontrivial facets that pass through it, construct complete subgraphs of the graph of $P(G,g_0),$ and show that these polyhedra are of diameter 2.