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 ×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.

The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [4] by introducing a vast hierarchy of generalizations to the notion of graph diameter. This hierarchy provides some interesting lower bounds for the usual graph diameter. After explaining the structure of the hierarchy and discussing these bounds, we focus on clearly explaining the differences and similarities among the many diameter notions of our hierarchy. Finally, we fully characterize the hierarchy in dimension two. It collapses into fewer categories, for which we exhibit the ranges of values that can be realized as diameters.MSC 2010: 52B05, 52B55, 52B40, 52C40, 52C45, 90C05, 90C49.

Abstract. We solve a problem in the combinatorics of polyhedra motivated by the network simplex method. We show that the Hirsch conjecture holds for the diameter of the graphs of all network-flow polytopes, in particular the diameter of a network-flow polytope for a network with n nodes and m arcs is never more than m + n − 1. A key step to prove this is to show the same result for classical transportation polytopes.

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The study of the diameter of the graph of polyhedra is a classical problem in the theory of linear programming. While transportation polytopes are at the core of operations research and statistics it is still open whether the Hirsch conjecture is true for general m×n-transportation polytopes. In earlier work the first three authors introduced a hierarchy of variations to the notion of graph diameter in polyhedra. The key reason was that this hierarchy provides some interesting lower bounds for the usual graph diameter. This paper has three contributions: First, we compare the hierarchy of diameters for the m×n-transportation polytopes. We show that the Hirsch conjecture bound of m + n − 1 is actually valid in most of these diameter notions. Second, we prove that for 3×n-transportation polytopes the Hirsch conjecture holds in the classical graph diameter. Third, we show for 2×n-transportation polytopes that the stronger monotone Hirsch conjecture holds and improve earlier bounds on the graph diameter.