(Almost) tight bounds and existence theorems for confluent flows

Jiangzhuo Chen, Robert D. Kleinberg, L. Lovász, Rajmohan Rajaraman, Ravi Sundaram, Adrian Vetta

Research output: Chapter in Book/Report/Conference proceedingConference contribution

8 Citations (Scopus)

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 languageEnglish
Title of host publicationConference Proceedings of the Annual ACM Symposium on Theory of Computing
Pages529-538
Number of pages10
Publication statusPublished - 2004
EventProceedings of the 36th Annual ACM Symposium on Theory of Computing - Chicago, IL, United States
Duration: Jun 13 2004Jun 15 2004

Other

OtherProceedings of the 36th Annual ACM Symposium on Theory of Computing
CountryUnited States
CityChicago, IL
Period6/13/046/15/04

Fingerprint

Polynomials
Internet
Directed graphs
Approximation algorithms
Hardness

Keywords

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

ASJC Scopus subject areas

  • Software

Cite this

Chen, J., Kleinberg, R. D., Lovász, L., Rajaraman, R., Sundaram, R., & Vetta, A. (2004). (Almost) tight bounds and existence theorems for confluent flows. In 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.

Conference Proceedings of the Annual ACM Symposium on Theory of Computing. 2004. p. 529-538.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Chen, J, Kleinberg, RD, Lovász, L, Rajaraman, R, Sundaram, R & Vetta, A 2004, (Almost) tight bounds and existence theorems for confluent flows. in 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.
Chen J, Kleinberg RD, Lovász L, Rajaraman R, Sundaram R, Vetta A. (Almost) tight bounds and existence theorems for confluent flows. In Conference Proceedings of the Annual ACM Symposium on Theory of Computing. 2004. p. 529-538
Chen, Jiangzhuo ; Kleinberg, Robert D. ; Lovász, L. ; Rajaraman, Rajmohan ; Sundaram, Ravi ; Vetta, Adrian. / (Almost) tight bounds and existence theorems for confluent flows. Conference Proceedings of the Annual ACM Symposium on Theory of Computing. 2004. pp. 529-538
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