 
Summary: AN O(log k) APPROXIMATE MINCUT MAXFLOW 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 kcommodity flow instances with arbitrary capacities and demands. This improves
upon the previously bestknown 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 mincut ratio, is presented.
Key words. approximation algorithms, cuts, sparse cuts, network flow, multicommodity flow
AMS subject classifications. 05C38, 68R10, 90B10
PII. S0097539794285983
1. Introduction.
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
