Summary: AN O(log k) APPROXIMATE MIN-CUT MAX-FLOW THEOREM
AND APPROXIMATION ALGORITHM
YONATAN AUMANN AND YUVAL RABANI
SIAM J. COMPUT. c 1998 Society for Industrial and Applied Mathematics
Vol. 27, No. 1, pp. 291301, February 1998 012
Abstract. It is shown that the minimum cut ratio is within a factor of O(log k) of the maximum
concurrent flow for k-commodity flow instances with arbitrary capacities and demands. This improves
upon the previously best-known bound of O(log2
k) and is existentially tight, up to a constant factor.
An algorithm for finding a cut with ratio within a factor of O(log k) of the maximum concurrent
flow, and thus of the optimal min-cut ratio, is presented.
Key words. approximation algorithms, cuts, sparse cuts, network flow, multicommodity flow
AMS subject classifications. 05C38, 68R10, 90B10
1.1. Multicommodity flow. Consider an undirected graph G = (V, E) with
an assignment of nonnegative capacities to the edges, c : E R+
. A multicommodity
flow instance on G is a set of ordered pairs of vertices (s1, t1), (s2, t2), . . . , (sk, tk).
Each pair (si, ti) represents a commodity, with source at si and destination or target