# 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} }

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…

## 30 Citations

Maximum edge-disjoint paths in planar graphs with congestion 2

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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.

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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.

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

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- 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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- 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.

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

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- 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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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.

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- Computer Science, Mathematics2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
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The first deterministic, almost-linear time approximation algorithm for the classical Minimum Balanced Cut problem, which provides a stronger guarantee: it either returns a balanced cut whose value is close to a given target value, or it certifies that such a cut does not exist by exhibiting a large subgraph of $G$ that has high conductance.

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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.

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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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