### (Almost) Tight bounds and existence theorems for single-commodity confluent flows(Almost) Tight bounds and existence theorems for single-commodity confluent flows Access Restriction
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 Author Chen, Jiangzhuo ♦ Kleinberg, Robert D. ♦ Lovsz, Lszl ♦ Rajaraman, Rajmohan ♦ Sundaram, Ravi ♦ Vetta, Adrian Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2007 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Approximation algorithms ♦ Confluent flow ♦ Network flow ♦ Routing ♦ Tight bounds 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. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2007-07-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 54 Issue Number 4 Page Count 32 Starting Page 1 Ending Page 32

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Source: ACM Digital Library