On the circuit diameter of dual transportation polyhedra

TL;DR

This paper introduces the concept of circuit diameter for polyhedra, establishing a new upper bound that is tighter than the known combinatorial diameter bounds for dual transportation polyhedra on bipartite graphs.

Contribution

It defines the circuit diameter and proves a stronger upper bound for dual transportation polyhedra on arbitrary bipartite graphs, extending previous bounds.

Findings

Circuit diameter is always bounded above by the combinatorial diameter.

For complete bipartite graphs, the known Hirsch bound is tight.

A new upper bound of M+N-2 is established for all bipartite graphs.

Abstract

In this paper we introduce the circuit diameter of polyhedra, which is always bounded from above by the combinatorial diameter. We consider dual transportation polyhedra defined on general bipartite graphs. For complete $M{\times}N$ bipartite graphs the Hirsch bound $(M{-}1)(N{-}1)$ on the combinatorial diameter is a known tight bound (Balinski, 1984). For the circuit diameter we show the much stronger bound $M{+}N{-}2$ for all dual transportation polyhedra defined on arbitrary bipartite graphs with $M{+}N$ nodes.