A polynomial bound on the diameter of the transportation polytope

We prove that the combinatorial diameter of the skeleton of the polytope of feasible solutions to any m×n transportation problem is O(max{n, m}). The transportation problem (TP) is a classic problem in operations research. The problem was posed for the first time by Hitchcock in 1941 [8] and independently by Koopmans in 1947 [11]. The problem 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 ri units, and each demand point j wants cj units. A solution to the problem can be written as a m×n matrix X with entries decision variables xij having value equal to the number of units transported from supply point i to demand point j. Without loss of generality we assume that ∑m i=1 ri = ∑n j=1 cj . The objective is to minimize total transportation costs ∑m i=1 ∑n j=1 tijxij , where tij is the unit transportation cost from supply point i to demand point j. The set of feasible solutions of TP, the transportation polytope, is described by