13 November 2007

13:30

Leen Stougie

Abstract

The transportation problem (TP) is a classic problem in operations research. The problem was posed for the first time by Hitchcock in 1941 [Hitchcock, 1941] and independently by Koopmans in 1947 [Koopmans, 1948], and appears in any standard introductory course on operations research.
The mxn TP has m supply points and n demand points. Each supply Point i holds a quantity r_i, and each demand point j wants a quantity c_j, with the sum of femands equal to the sum of supplies. A solution to the problem can be written as a mxn matrix X with entries decision x_{ij} having value equal to the amount transported from supply point i to demand point j. The objective is to minimize total transportation costs when unit transporation costs between each supply and each demand point are given.
The set of feasible solutions of TP, is called the transportation polytope.
The 1-skeleton (edge graph) of this polytope is defined as the graph with vertices the vertices of the polytope and edges its 1-dimensional faces.
In 1957 W.M. Hirsch stated his famous conjecture cf. [Dantzig, 1963]) saying that any d-dimensional polytope with n facets has diameter at most n-d. So far the best bound for any polytope is O(n^{\log d+1}) [Kalai and Kleitman, 1992]. Any strongly polynomial bound is still lacking. Such bounds have been proved for some special classes of polytopes (for examples, see [Schrijver, 1995]). Among those are some special classes of transportation polytopes [Balinski, 1974],[Bolker, 1972] and the polytope of the dual of TP [Balinski, 1974].
The first strongly polynomial bound on the diameter of the transportation polytope was given by Dyer and Frieze [DyerFrieze, 1994]. Actually, they prove a bound on the diameter of any polytope {x|Ax=b} where A is a totally unimodular matrix. The proof is complicated and indirect, using the probabilistic method. Moreover, the bound is huge O(m^{16}n^3ln(mn))3) assuming m less than or equal to n.
We will give a simple proof that the diameter of the transportation polytope is less than 8(m+n-2). The proof is constructive: it gives an algorithm that describes how to go from any vertex to any other vertex on the transportation polytope in less than 8(m+n-2) steps along the edges.
According to the Hirsch Conjecture the bound on the TP polytope should be
m+n-1. Thus we are within a multiplicative factor 8 of the Hirsch bound.
Recently C. Hurkens refined our analysis and diminished the bound by a factor 2, arriving at 4(m+n-2). I will indicate the way he achieved this as well.