Faster Algorithms for the Quickest Transshipment Problem

A transshipment problem with demands that exceed network capacity can be solved by sending flow 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 [ Math. Oper. Res., 25 (2000), pp. 36--62] describe the only known polynomial time algorithm to solve this problem. They actually solve the significantly harder problem in which it takes a prespecified amount of time for flow to travel from one end of an arc to the other. Their algorithm repeatedly calls an oracle for submodular function minimization. We present an algorithm that finds a quickest transshipment with a polynomial number of maximum flow computations, and a faster algorithm that also uses minimum cost flow computations. When there is only one sink, we show how the algorithm can be sped up to return a solution using O(k) maximum flow computations, where k is the number of sources. Hajek and Ogier [Networks, 14 (1984), pp. 457--487] describe an algorithm that finds a fractional solution to the single sink quickest transshipment problem on a network with n nodes and m arcs using O(n) maximum flow computations. They actually solve the universally quickest transshipment---a flow over time that minimizes the amount of supply left in the network at every moment of time. In this paper, we show how to solve the universally quickest transshipment in O(mnlog(n2/m)) time, the same asymptotic time as a push-relabel maximum flow computation.