# The Quickest Multicommodity Flow Problem

## Abstract

Traditionally, flows over time are solved in time-expanded networks which contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static flows, its main and often fatal drawback is the enormous size of the time-expanded network. In particular, this approach usually does not lead to efficient algorithms with running time polynomial in the input size since the size of the time-expanded network is only pseudo-polynomial.

We present two different approaches for coping with this difficulty. Firstly, inspired by the work of Ford and Fulkerson on maximal *s-t*-flows over time (or ‘maximal dynamic *s-t*-flows’), we show that static, length-bounded flows lead to provably good multicommodity flows over time. These solutions not only feature a simple structure but can also be computed very efficiently in polynomial time.

Secondly, we investigate ‘condensed’ time-expanded networks which rely on a rougher discretization of time. Unfortunately, there is a natural tradeoff between the roughness of the discretization and the quality of the achievable solutions. However, we prove that a solution of arbitrary precision can be computed in polynomial time through an appropriate discretization leading to a condensed time expanded network of polynomial size. In particular, this approach yields a fully polynomial time approximation scheme for the quickest multicommodity flow problem and also for more general problems.

## Keywords

Polynomial Time Transit Time Intermediate Node Discrete Time Model Input Size## Preview

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