### Abstract

A flow is said to be confluent if at any node all the flow leaves along a single edge. Given a directed graph G with k sinks arid non-negative demands on all the nodes of G, we consider the problem of determining a confluent flow that routes every node demand to some sink such that the maximum congestion at a sink is minimized. Confluent flows arise in a variety of application areas, most notably in networking; in fact, most flows in the Internet are confluent since Internet routing is destination based. We present near-tight approximation algorithms, hardness results, and existence theorems for confluent flows. The main result of this paper is a polynomial-time algorithm for determining a confluent flow with congestion at most 1 + ln(k) in G, if G admits a splittable flow with congestion at most 1. We complement this result in two directions. First, we present a graph G that admits a splittable flow with congestion at most 1, yet no confluent flow with congestion smaller than H _{k}, thus establishing tight upper and lower bounds to within an additive constant less than 1. Second, we show that it is NP-hard to approximate the con gestion of an optimal confluent flow to within a factor of (lg k)/2, thus resolving the polynomial-time approximability to within a multiplicative constant. We also consider a demand maximization version of the problem. We show that if G admits a splittable flow of congestion at most 1, then a variant of the congestion minimization algorithm yields a confluent flow in G with congestion at most 1 that satisfies 1/3 fraction of total demand. We show that the gap between confluent flows and splittable flows is much smaller, if the underlying graph were k-connected. In particular, we prove that k-connected graphs with k sinks admit confluent flows of congestion less than C + d _{max}, where C is the congestion of the best splittable flow, and d _{max} is the maximum demand of any node in G. The proof of this existence theorem is non-constructive and relies on topological techniques introduced in [16].

Original language | English |
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Title of host publication | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

Pages | 529-538 |

Number of pages | 10 |

Publication status | Published - 2004 |

Event | Proceedings of the 36th Annual ACM Symposium on Theory of Computing - Chicago, IL, United States Duration: Jun 13 2004 → Jun 15 2004 |

### Other

Other | Proceedings of the 36th Annual ACM Symposium on Theory of Computing |
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Country | United States |

City | Chicago, IL |

Period | 6/13/04 → 6/15/04 |

### Fingerprint

### Keywords

- Approximation algorithms
- Confluent flow
- Network flow
- Routing
- Tight bounds

### ASJC Scopus subject areas

- Software

### Cite this

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing*(pp. 529-538)

**(Almost) tight bounds and existence theorems for confluent flows.** / Chen, Jiangzhuo; Kleinberg, Robert D.; Lovász, L.; Rajaraman, Rajmohan; Sundaram, Ravi; Vetta, Adrian.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing.*pp. 529-538, Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, United States, 6/13/04.

}

TY - GEN

T1 - (Almost) tight bounds and existence theorems for confluent flows

AU - Chen, Jiangzhuo

AU - Kleinberg, Robert D.

AU - Lovász, L.

AU - Rajaraman, Rajmohan

AU - Sundaram, Ravi

AU - Vetta, Adrian

PY - 2004

Y1 - 2004

N2 - A flow is said to be confluent if at any node all the flow leaves along a single edge. Given a directed graph G with k sinks arid non-negative demands on all the nodes of G, we consider the problem of determining a confluent flow that routes every node demand to some sink such that the maximum congestion at a sink is minimized. Confluent flows arise in a variety of application areas, most notably in networking; in fact, most flows in the Internet are confluent since Internet routing is destination based. We present near-tight approximation algorithms, hardness results, and existence theorems for confluent flows. The main result of this paper is a polynomial-time algorithm for determining a confluent flow with congestion at most 1 + ln(k) in G, if G admits a splittable flow with congestion at most 1. We complement this result in two directions. First, we present a graph G that admits a splittable flow with congestion at most 1, yet no confluent flow with congestion smaller than H k, thus establishing tight upper and lower bounds to within an additive constant less than 1. Second, we show that it is NP-hard to approximate the con gestion of an optimal confluent flow to within a factor of (lg k)/2, thus resolving the polynomial-time approximability to within a multiplicative constant. We also consider a demand maximization version of the problem. We show that if G admits a splittable flow of congestion at most 1, then a variant of the congestion minimization algorithm yields a confluent flow in G with congestion at most 1 that satisfies 1/3 fraction of total demand. We show that the gap between confluent flows and splittable flows is much smaller, if the underlying graph were k-connected. In particular, we prove that k-connected graphs with k sinks admit confluent flows of congestion less than C + d max, where C is the congestion of the best splittable flow, and d max is the maximum demand of any node in G. The proof of this existence theorem is non-constructive and relies on topological techniques introduced in [16].

AB - A flow is said to be confluent if at any node all the flow leaves along a single edge. Given a directed graph G with k sinks arid non-negative demands on all the nodes of G, we consider the problem of determining a confluent flow that routes every node demand to some sink such that the maximum congestion at a sink is minimized. Confluent flows arise in a variety of application areas, most notably in networking; in fact, most flows in the Internet are confluent since Internet routing is destination based. We present near-tight approximation algorithms, hardness results, and existence theorems for confluent flows. The main result of this paper is a polynomial-time algorithm for determining a confluent flow with congestion at most 1 + ln(k) in G, if G admits a splittable flow with congestion at most 1. We complement this result in two directions. First, we present a graph G that admits a splittable flow with congestion at most 1, yet no confluent flow with congestion smaller than H k, thus establishing tight upper and lower bounds to within an additive constant less than 1. Second, we show that it is NP-hard to approximate the con gestion of an optimal confluent flow to within a factor of (lg k)/2, thus resolving the polynomial-time approximability to within a multiplicative constant. We also consider a demand maximization version of the problem. We show that if G admits a splittable flow of congestion at most 1, then a variant of the congestion minimization algorithm yields a confluent flow in G with congestion at most 1 that satisfies 1/3 fraction of total demand. We show that the gap between confluent flows and splittable flows is much smaller, if the underlying graph were k-connected. In particular, we prove that k-connected graphs with k sinks admit confluent flows of congestion less than C + d max, where C is the congestion of the best splittable flow, and d max is the maximum demand of any node in G. The proof of this existence theorem is non-constructive and relies on topological techniques introduced in [16].

KW - Approximation algorithms

KW - Confluent flow

KW - Network flow

KW - Routing

KW - Tight bounds

UR - http://www.scopus.com/inward/record.url?scp=4544347147&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4544347147&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:4544347147

SP - 529

EP - 538

BT - Conference Proceedings of the Annual ACM Symposium on Theory of Computing

ER -