# An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations

@inproceedings{Kelner2014AnAA, title={An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations}, author={Jonathan A. Kelner and Lorenzo Orecchia and Yin Tat Lee and Aaron Sidford}, booktitle={SODA}, year={2014} }

In this paper, we introduce a new framework for approximately solving flow problems in capacitated, undirected graphs and apply it to provide asymptotically faster algorithms for the maximum s-t flow and maximum concurrent multicommodity flow problems. For graphs with n vertices and m edges, it allows us to find an e-approximate maximum s-t flow in time O(m1+o(1)e-2), improving on the previous best bound of O(mn1/3poly(e-1)). Applying the same framework in the multicommodity setting solves a…

## 247 Citations

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An algorithm which given any m-edge n-vertex directed graph with integer capacities at most U computes a maximum s-t flow for any vertices s and t in m 11/8+o(1) U 1/4 time with high probability.

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- Computer Science2013 IEEE 54th Annual Symposium on Foundations of Computer Science
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The notion of balance for directed graphs is introduced and an efficient algorithm that finds an (1+є)-approximate maximum flows in α-balanced graphs in time O(m α2 / є2) is developed, which can efficiently determine whether a given directed graph is α- balanced.

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