An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms

A multicommodity flow problem is considered where for each pair of vertices (u, v) it is required to send f half-units of commodity (u, v) from u to v and f half-units of commodity (v, u) from v to u without violating capacity constraints. The main result is an algorithm for performing the task provided that the capacity of each cut exceeds the demand across the cut by a Θ(log n) factor. The condition on cuts is required in the worst case, and is trivially within a Θ(log n) factor of optimal for any flow problem. The result can be used to construct the first polylog-times optimal approximation algorithms for a wide variety of problems, including minimum quotient separators, 1/3-2/3 separators, bifurcators, crossing number, and VLSI layout area. It can also be used to route packets efficiently in arbitrary distributed networks