# Tech Report CS-93-30

## Bounds on the Max-Flow Min-Cut Ratio for Directed Multicommodity Flows

### Philip N. Klein, Serge A. Plotkin, Satish Rao and Eva Tardos

#### September 1993

### Abstract:

The best-known theorem in combinatorial optimization is the classical max-flow min-cut theorem of Ford and Fulkerson. This theorem serves as the basis for deriving efficient algorithms for finding max-flows and min-cuts. Starting with the work of Leighton and Rao, significant effort was directed towards finding approximate analogs for the {\em undirected} multicommodity flow problem.

In this paper we consider an approximate max-flow min-cut theorem for {\em directed} graphs. We prove a polylogarithmic bound on the worst case ratio between the minimum multicut and the value of the maximum multicommodity flow in the special case when the demands are symmetric. The method presented in this paper can be used to give polynomial time polylogarithmic approximation algorithms for the corresponding minimum directed multicut problems.

The problem with symmetric demands extends the only previously known special case concerning directed graphs due to Leighton and Rao, who proved an $O(\log n)$ bound for the case when there is a unit demand between every pair of nodes. Computation of minimum cuts in directed multicommodity flow problems with symmetric demand is a basic step for approximation algorithms for a number of NP-complete problems. As an example, we show how to use our multicut approximation algorithm to approximately solve the minimum clause-deletion problem for 2-CNF formulae.

We also consider a generalization of the minimum-cut problem where instead of separating pairs of terminals, we have to separate sets of terminals. Our main result is a polynomial-time polylogarithmic approximation algorithm for this generalized minimum-cut problem.

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