# 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}
}
• M. Andrews
• Published 23 October 2010
• Computer Science, Mathematics
• 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
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…

## Figures from this paper

A Polylogarithmic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2
• Computer Science, Mathematics
FOCS
• 2012
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.
Improved Approximation for Node-Disjoint Paths in Grids with Sources on the Boundary
• Computer Science, Mathematics
ICALP
• 2018
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, Mathematics
2012 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.
Maximum edge-disjoint paths in planar graphs with congestion 2
• Mathematics
Math. Program.
• 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
• Mathematics
SODA
• 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.
On Approximating Node-Disjoint Paths in Grids
• Computer Science, Mathematics
APPROX-RANDOM
• 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 Science
ACM 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.
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.
New hardness results for routing on disjoint paths
• Computer Science, Mathematics
STOC
• 2017
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
• Computer Science, Mathematics
STOC
• 2016
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.

## References

SHOWING 1-10 OF 40 REFERENCES
Approximating Disjoint-Path Problems Using Greedy Algorithms and Packing Integer Programs
• Computer Science, Mathematics
IPCO
• 1998
These techniques lead to the first approximation algorithm and obtain an approximation ratio that matches, to within logarithmic factors, the $O(\sqrt{|E|})$ approximation ratio for the simple edge-disjoint path problem.
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 routing with congestion in directed graphs
• Computer Science
STOC '07
• 2007
The hardness construction with the perfect completeness restriction allows us to conclude that the directedcongestion minimization problem, where the goal is to route all pairs with minimum congestion, is hard to approximate to within afactor of Ω(log N/log log N).
Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems
• Computer Science, Mathematics
STOC '99
• 1999
It is shown that in directed networks, for any > 0, EDP is NP-hard to approximate within m 1=2 and simple approximation algorithms are designed that achieve essentially matching approximation guarantees for some generalizations of EDP.
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.
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.
Hardness of the Undirected Edge-Disjoint Paths Problem with Congestion
• Computer Science
FOCS
• 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.
Existence and Construction of Edge-Disjoint Paths on Expander Graphs
• Mathematics, Computer Science
SIAM J. Comput.
• 1994
The authors prove sufficient conditions for the existence of edge-disjoint paths connecting any set of $q\leq n/(\log n)^\kappa$ disjoint pairs of vertices on any $n$ vertex bounded degree expander, where $\ kappa$ depends only on the expansion properties of the input graph, and not on $n$.
Short paths in expander graphs
• Computer Science
Proceedings of 37th Conference on Foundations of Computer Science
• 1996
This work shows that a greedy algorithm for approximating the maximum disjoint paths problem achieves a polylogarithmic approximation ratio in bounded-degree expanders, and develops new routing algorithms and structural results for bounded- degree expander graphs.
An O(sqrt(n)) Approximation and Integrality Gap for Disjoint Paths and Unsplittable Flow
• Computer Science, Mathematics
Theory Comput.
• 2006
In undirected graphs and directed acyclic graphs, the maximization version of the edge-disjoint path problem (EDP), an O( p n) upper bound on the approximation ratio where n is the number of nodes in the graph is obtained.