We prove that the combinatorial diameter of the skeleton of the polytope of feasible solutions of any m × n transportation problem is at most 8(m + n − 2). The transportation problem ( TP) is a classic problem in operations research. The problem was posed for the first time by Hitchcock in 1941 [9] and independently by Koopmans in 1947 [12], and appears in any standard introductory course on operations research. The m × n TP has m supply points and n demand points. Each supply point i holds a quantity ri> 0, and each demand point j wants a quantity cj> 0, with m� ri = n� cj. A solution to the problem can be written as an m × n matrix X, where entries are decision variables xij having value equal to the amount transported from supply point i to demand point j. The set of feasible solutions of TP, the transportation polytope T, is described by n� xij = ri, i = 1, 2,...,m; j=1 m� xij = cj, j = 1, 2,...,n; i=1 xij ≥ 0, i = 1, 2,...,m, j = 1, 2,...n. The 1-skeleton ( edge graph) of T is defined as the graph with vertices the vertices of the polytope and edges its 1-dimensional faces. The diameter of T, which we denote by diam(T), is the diameter of its 1-skeleton. In 1957 W.M. Hirsch stated his famous conjecture (cf. [5]) saying that any d-dimensional polytope with n facets has diameter at most n − d. So far the best known bound for arbitrary polytopes is O(n log d+1) [10]. Any polynomial bound is still lacking. Such bounds have been proved for some special classes of polytopes ( for examples, see [14]). Among those are some special classes of transportation polytopes [1, 3] and the polytope of the dual of TP [1].

Abstract. This paper describes the economical placing of diagonal braces in the walls and ceiling of a rectangularone story building. It beginswith the definition ofthe structuregeometry of agraphembedded in Euclidean space: a combinatorial geometry (matroid) on the set of potential braces. When the embedded graph is a plane grid of squares thegeometry is graphic. Then, for example, minimal rigidifying sets of braces correspond to spanning trees in a complete bipartite graph. The methods used in the plane case are extended to analyze how sets of wall and ceiling braces in a one story building can be dependent. 1. Introduction. Interest

We prove that the combinatorial diameter of the skeleton of the polytope of feasible solutions of any m × n transportation problem is less than 1/2 (m + n)².

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