# Approximation Algorithms for the Edge-Disjoint Paths Problem via Raecke Decompositions

@article{Andrews2010ApproximationAF, title={Approximation Algorithms for the Edge-Disjoint Paths Problem via Raecke Decompositions}, author={Matthew Andrews}, journal={2010 IEEE 51st Annual Symposium on Foundations of Computer Science}, year={2010}, pages={277-286} }

We study the Edge-Disjoint Paths with Congestion (EDPwC) problem in undirected networks in which we must integrally route a set of demands without causing large congestion on an edge. We present 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 $poly(\log\log n)$. The best previous…

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## 31 Citations

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

- Computer Science, MathematicsFOCS
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An O(\poly\log k)-approximation algorithm is shown that gives the best possible congestion for a sub-polynomial approximation of \EDPwC via this relaxation, close to optimal in terms of the number of pairs routed.

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- Computer Science, MathematicsICALP
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This work uses a (completely different) linear program only to select the pairs to be routed, while the routing itself is computed by other methods, resulting in an efficient randomized $2^{O(\sqrt{\log n} \cdot \log\log n)}$-approximation algorithm for this problem.

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

- Computer Science, Mathematics2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
- 2012

An O(poly log k)-approximation algorithm is shown for EDPwC with congestion c = 2, by rounding the standard multicommodity How relaxation of the problem, which gives the best possible congestion for a sub-polynomial approximation of EDP wC via this relaxation.

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- 2021

If every edge has capacity at least 2, then the integrality gap drops to a constant for planar graphs, and the concept of rooted clustering is introduced which the author believes is of independent interest.

Poly-logarithmic Approximation for Maximum Node Disjoint Paths with Constant Congestion

- MathematicsSODA
- 2013

This work gives a polynomial time algorithm that routes Ω(OPT/poly log k) pairs with O(1) congestion, where OPT is the value of an optimum fractional solution to a natural multicommodity flow relaxation.

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- 2015

The algorithm shows that when all demand pairs are of the latter type, the integrality gap of the multicommodity flow LP-relaxation is at most O(n^{1/4} * log(n), and it is complemented by proving that NDP is APX-hard on grid graphs.

An Improved Approximation Algorithm for the Edge-Disjoint Paths Problem with Congestion Two

- Mathematics, Computer ScienceACM Trans. Algorithms
- 2016

This article gives a randomized O(n3/7 ċ poly(log n))-approximation algorithm with congestion two, and proves that there is a (randomized) polynomial-time algorithm for finding Ω(OPT1/p) edge-disjoint paths connecting given terminal pairs for some p > 1.

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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.

New hardness results for routing on disjoint paths

- Computer Science, MathematicsSTOC
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It is proved that NDP is 2Ω(√logn)-hard to approximate, unless all problems in NP have algorithms with running time nO(logn), and this result holds even when the underlying graph is a planar graph with maximum vertex degree 4, and all source vertices lie on the boundary of a single face.

Improved approximation for node-disjoint paths in planar graphs

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A new linear programming relaxation of the classical Node-Disjoint Paths problem is introduced, and a number of new techniques are introduced that are hoped will be helpful in designing more powerful algorithms for this and related problems.

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