# A Polylogarithmic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2

@article{Chuzhoy2012APA,
title={A Polylogarithmic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2},
author={Julia Chuzhoy and Shi Li},
journal={ArXiv},
year={2012},
volume={abs/1208.1272}
}
• Published 6 August 2012
• Computer Science, Mathematics
• ArXiv
In the Edge-Disjoint Paths with Congestion problem (\EDPwC), we are given an undirected $n$-vertex graph $G$, a collection $\mset=\set{(s_1, t_1), \ldots, (s_k, t_k)}$ of demand pairs and an integer $c$. The goal is to connect the maximum possible number of the demand pairs by paths, so that the maximum edge congestion - the number of paths sharing any edge - is bounded by $c$. When the maximum allowed congestion is $c=1$, this is the classical Edge-Disjoint Paths problem (\EDP). The best…
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## References

SHOWING 1-10 OF 32 REFERENCES
Approximation Algorithms for the Edge-Disjoint Paths Problem via Raecke Decompositions
• M. Andrews
• Computer Science, Mathematics
2010 IEEE 51st Annual Symposium on Foundations of Computer Science
• 2010
This work presents a $(polylog(n), poly(\log\log n))-approximation, which means that if there exists a solution that routes X demands integrally on edge-disjoint paths (i.e. with congestion$1), then the approximation algorithm can route X/polylog (n) demands with congestion.
Inapproximability of Edge-Disjoint Paths and low congestion routing on undirected graphs
• Mathematics, Computer Science
Comb.
• 2007
This paper studies the hardness of EDPwC in undirected graphs and shows an Ω(log logV/log log log V) hardness of approximation for EDPWC and an Φ(loglogV/ log log logV) hardness-of- approximation for the Undirected congestion minimization problem.
Optimal construction of edge-disjoint paths in random graphs
• Mathematics, Computer Science
SODA '94
• 1994
The results give the first tight bounds for the edge-disjoint paths problem for any nontrivial class of graphs.
Routing in undirected graphs with constant congestion
An efficient randomized algorithm is shown that routes Ω(OPT/poly log k) demand pairs with congestion at most 14, where OPT is the maximum number of pairs that can be simultaneously routed on edge-disjoint paths.
Breaking o(n1/2)-approximation algorithms for the edge-disjoint paths problem with congestion two
• Mathematics, Computer Science
STOC '11
• 2011
This paper gives a randomized O(n3/7 • poly (log n))-approximation algorithm with congestion two, and shows that there is a randomized algorithm for finding Ω(OPT1/4) edge-disjoint paths connecting given terminal pairs with congestionTwo.
Edge Disjoint Paths in Moderately Connected Graphs
• Mathematics, Computer Science
SIAM J. Comput.
• 2010
This work shows a polylogarithmic approximation algorithm for the undirected EDP problem in general graphs with a moderate restriction on graph connectivity, and extends previous techniques in that it applies to graphs with high diameters and asymptotically large minors.
Hardness of the undirected edge-disjoint paths problem with congestion
• Computer Science
46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)
• 2005
An improved hardness result for EDP is obtained, and the first polylogarithmic integrality gaps and hardness of approximation results for E DPwC are shown, and similar results can be obtained for the all-or-nothing flow (ANF) problem, a relaxation of EDP.
Edge-disjoint paths in expander graphs
• A. Frieze
• Mathematics, Computer Science
SODA '00
• 2000
It is shown that if G has sufficiently strong expansion properties and r is sufficiently large, then all sets of $\kappa=\Omega(n/\log n)$ pairs of vertices can be joined.
Edge-disjoint paths in Planar graphs with constant congestion
• Mathematics, Computer Science
STOC '06
• 2006
A constant factor approximation for the all-or-nothing flow problem on OS instances via the flow relaxation is developed and a lower bound of Ω(log n) is shown for general graphs and for planar graphs that suggest a super-constant lower bound.
Edge-disjoint paths in planar graphs
• Mathematics
45th Annual IEEE Symposium on Foundations of Computer Science
• 2004
The heart of the approach is to show that in any undirected planar graph, given any matching M on a well-linked set X, the authors can route /spl Omega/(|M|) pairs in M with a congestion of 2, and this result extends to the unsplittable flow problem and the maximum integer multicommodity flow problem.