Faster Algorithms for the Quickest Transshipment Problem

A transshipment problem with demands that exceed network capacity can be solved by sending ow in several waves. How can this be done in the minimum number of waves? This is the question tackled in the quickest transshipment problem. Hoppe and Tardos 10] describe the only known polynomial time algorithm to solve this problem. Their algorithm repeatedly minimizes submodular functions using the ellipsoid method, and is therefore not at all practical. We present an algorithm that nds a quickest transshipment with a polynomial number of maximum ow computations, and a faster algorithm that also uses minimum cost ow computations. When there is only one sink, we show how the algorithm can be sped up to return a solution using O(k) maximum ow computations, where k is the number of sources. Hajek and Ogier 9] describe an algorithm that nds a fractional solution to the single-sink quickest transshipment problem on a network with n nodes using O(n) maximum ow computations. They actually solve the universally quickest transshipment|a dynamic ow that minimizes the amount of supply left in the network at every moment of time. In this paper, we show how to solve this problem in O(mn log(n 2 =m)) time, the same asymptotic time required by the fastest known algorithm to compute a maximum ow.