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

Jiangzhuo Chen, Robert D. Kleinberg, László Lovász, Rajmohan Rajaraman, Ravi Sundaram, Adrian Vetta
2007 Journal of the ACM
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 paper 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
more » ... sed. 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 G with k sinks and non-negative demands on all the nodes of 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 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 , the 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 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 is 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 by Lovász .