Approximate minimum-cost multicommodity flows in$\tilde O$(ɛ ⁻² KNM) timetime

Abstract

We show that an ε-approximate solution of the cost-constrainedK-commodity flow problem on anN-nodeM-arc network,G can be computed by sequentially solving O(K(ɛ ⁻²+logGK) logGM log (Gɛ ⁻¹ GK)) single-commodity minimum-cost flow problems on the same network. In particular, an approximate minimum-cost multicommodity flow can be computed in$\tilde O$(Gɛ ⁻² GKNM) running time, where the notation Õ(·) means “up to logarithmic factors”. This result improves the time bound mentioned by Grigoriadis and Khachiyan [4] by a factor ofM/N and that developed more recently by Karger and Plotkin [8] by a factor ofɛ ⁻¹. We also provide a simple$\tilde O$(NM)-time algorithm for single-commodity budget-constrained minimum-cost flows which is$\tilde O$(ɛ ⁻³) times faster than the algorithm developed in the latter paper.