A fast parametric maximum flow algorithm and applications
SIAM Journal on Computing
Approximation algorithms for scheduling unrelated parallel machines
Mathematical Programming: Series A and B
Designing multi-commodity flow trees
Information Processing Letters
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
Provisioning a virtual private network: a network design problem for multicommodity flow
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Algorithms for provisioning virtual private networks in the hose model
Proceedings of the 2001 conference on Applications, technologies, architectures, and protocols for computer communications
Hop-by-hop routing algorithms for premium traffic
ACM SIGCOMM Computer Communication Review
Meet and merge: approximation algorithms for confluent flows
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
How good can IP routing be?
Single-source unsplittable flow
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Journal of Computer and System Sciences
Energy efficient data gathering in multi-hop hierarchical wireless ad hoc networks
FOMC '11 Proceedings of the 7th ACM ACM SIGACT/SIGMOBILE International Workshop on Foundations of Mobile Computing
Capacitated confluent flows: complexity and algorithms
CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
Polynomial-time algorithms for special cases of the maximum confluent flow problem
Discrete Applied Mathematics
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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 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 article 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 Hk, the kth 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 (log2k)/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 + dmax, where C is the congestion of the best splittable flow, and dmax 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.