On the Combinatorial Diameters of Parallel and Series Connections

TL;DR

This paper studies how the combinatorial diameters of polyhedra formed from parallel and series connections of oriented matroids behave, showing they remain small under certain diameter bounds, advancing understanding of polyhedral diameter conjectures.

Contribution

It proves that the diameters of polyhedra from parallel or series connections stay within the Hirsch-conjecture bound if the original polyhedra do, aiding in the broader goal of bounding diameters of all totally-unimodular polyhedra.

Findings

Diameters of connected polyhedra remain small under the Hirsch bound.

Parallel connection adds a constant to the diameter bound.

Series connection's diameter depends on maximum coordinate value.

Abstract

The investigation of combinatorial diameters of polyhedra is a classical topic in linear programming due to its connection with the possibility of an efficient pivot rule for the simplex method. We are interested in the diameters of polyhedra formed from the so-called parallel or series connection of oriented matroids: oriented matroids are the natural way to connect representable matroid theory with the combinatorics of linear programming, and these connections are fundamental operations for the construction of more complicated matroids from elementary matroid blocks. We prove that, for polyhedra whose combinatorial diameter satisfies the Hirsch-conjecture bound regardless of the right-hand sides in a standard-form description, the diameters of their parallel or series connections remain small in the Hirsch-conjecture bound. These results are a substantial step toward devising a diameter bound for all polyhedra defined through totally-unimodular matrices based on Seymour's famous decomposition theorem. Our proof techniques and results exhibit a number of interesting features. While the parallel connection leads to a bound that adds just a constant, for the series connection one has to linearly take into account the maximal value in a specific coordinate of any vertex. Our proofs also require a careful treatment of non-revisiting edge walks in degenerate polyhedra, as well as the construction of edge walks that may take a `detour' to facets that satisfy the non-revisiting conjecture when the underlying polyhedron may not.