## Departmental Papers (CIS)

#### Date of this Version

June 2004

#### Document Type

Conference Paper

#### Recommended Citation

Chandra Chekuri, Sanjeev Khanna, and F. Bruce Shepherd, "The All-or-Nothing Multicommodity Flow Problem", . June 2004.

#### Abstract

We consider the all-or-nothing multicommodity flow problem in general graphs. We are given a capacitated undirected graph *G* = (*V, E, u*) and set of *k* pairs *s*_{1}*t*_{1}, *s*_{2}*t*_{2}, ..., *s*_{k}*t*_{k}. Each pair has a unit demand. The objective is to find a largest subset *S* of {1, 2, ..., *k*} such that for every *i* in *S* we can send a flow of one unit between *s _{i}* and

*t*. Note that this differs from the

_{i}*edge-disjoint path*problem (EDP) in that we do not insist on integral flows for the pairs. This problem is NP-hard, and APX-hard, even on trees. For trees, a 2-approximation is known for the cardinality case and a 4-approximation for the weighted case. In this paper we build on a recent result of Raecke on low congestion oblivious routing in undirected graphs to obtain a poly-logarithmic approximation for the all-or-nothing problem in

*general*undirected graphs. The best previous known approximation for all-or-nothing flow problem was

*O*(min(

*n*

^{2/3}, square root of

*m*)), the same as that for EDP. Our algorithm extends to the case where each pair

*s*has a demand

_{i}t_{i}*d*associated with it and we need to completely route

_{i}*d*to get credit for pair

_{i}*i*. We also consider the

*online admission control*version where pairs arrive online and the algorithm has to decide immediately on its arrival whether to accept it or not. We obtain a randomized algorithm with a competitive ratio that is similar to the approximation ratio for the offline algorithm.

#### Keywords

multicommodity flow, approximation algorithms, edge disjoint paths, oblivious routing, online algorithms

**Date Posted:** 24 September 2004

This document has been peer reviewed.

## Comments

Copyright ACM, 2004. This is the author's version of the work. It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in

Proceedings of the 36th Annual ACM Symposium on Theory of Computing 2004 (STOC 2004), pages 156-165.Publisher URL: http://doi.acm.org/10.1145/1007352.1007383