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

AbstractWe 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−1GK)) 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−2GKNM) 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.