NDLI: (Almost) Tight bounds and existence theorems for single-commodity confluent flows

Content Provider	ACM Digital Library
Author	Chen, Jiangzhuo Kleinberg, Robert D. Lovsz, Lszl Rajaraman, Rajmohan Sundaram, Ravi Vetta, Adrian
Copyright Year	2007
Abstract	A flow of a commodity is said to be confluent if at any node all the flow of the commodity leaves along a single edge. In this article, we study single-commodity confluent flow problems, where we need to route given node demands to a single destination using a confluent flow. Single- and multi-commodity confluent flows arise in a variety of application areas, most notably in networking; in fact, most flows in the Internet are (multi-commodity) confluent flows since Internet routing is destination based. We present near-tight approximation algorithms, hardness results, and existence theorems for minimizing congestion in single-commodity confluent flows. The maximum edge congestion of a single-commodity confluent flow occurs at one of the incoming edges of the destination. Therefore, finding a minimum-congestion confluent flow is equivalent to the following problem: given a directed graph $\textit{G}$ with $\textit{k}$ $\textit{sinks}$ and non-negative demands on all the nodes of $\textit{G},$ determine a confluent flow that routes every node demand to some sink such that the maximum congestion at a sink is minimized. The main result of this article is a polynomial-time algorithm for determining a confluent flow with congestion at most 1 + $ln(\textit{k})$ in $\textit{G},$ if $\textit{G}$ admits a splittable flow with congestion at most 1. We complement this result in two directions. First, we present a graph $\textit{G}$ that admits a splittable flow with congestion at most 1, yet no confluent flow with congestion smaller than $H_{k},$ the $\textit{k}th$ harmonic number, 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 congestion of an optimal confluent flow to within a factor of $(log_{2}k)/2,$ 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 $\textit{G}$ admits a splittable flow of congestion at most 1, then a variant of the congestion minimization algorithm yields a confluent flow in $\textit{G}$ 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 is $\textit{k}-connected.$ In particular, we prove that $\textit{k}-connected$ graphs with $\textit{k}$ sinks admit confluent flows of congestion less than $\textit{C}$ + $d_{max},$ where $\textit{C}$ is the congestion of the best splittable flow, and $d_{max}$ is the maximum demand of any node in $\textit{G}.$ The proof of this existence theorem is non-constructive and relies on topological techniques introduced by Lovász.
Starting Page	1
Ending Page	32
Page Count	32
File Format	PDF
ISSN	00045411
e-ISSN	1557735X
DOI	10.1145/1255443.1255444
Journal	Journal of the ACM (JACM)
Volume Number	54
Issue Number	4
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2007-07-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Approximation algorithms Confluent flow Network flow Routing Tight bounds
Content Type	Text
Resource Type	Article
Subject	Hardware and Architecture Information Systems Control and Systems Engineering Artificial Intelligence Software

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