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Breaking o(n^1/2)-approximation algorithms for the edge-disjoint paths problem with congestion two

Authors:

Ken-ichi Kawarabayashi,

Yusuke KobayashiAuthors Info & Claims

STOC '11: Proceedings of the forty-third annual ACM symposium on Theory of computing

Pages 81 - 88

https://doi.org/10.1145/1993636.1993648

Published: 06 June 2011 Publication History

Abstract

In the maximum edge-disjoint paths problem, we are given a graph and a collection of pairs of vertices, and the objective is to find the maximum number of pairs that can be routed by edge-disjoint paths. An r-approximation algorithm for this problem is a polynomial time algorithm that finds at least OPT / r edge-disjoint paths, where OPT is the maximum possible. Currently, an O(n^1/2)-approximation algorithm is best known for this problem even if a congestion of two is allowed, i.e., each edge is allowed to be used in at most two of the paths.

In this paper, we give a randomized O(n^3/7 • poly (log n))-approximation algorithm with congestion two. This is the first result that breaks the O(n^1/2)-approximation algorithm. In particular, we prove the following. 1. If we have a (randomized) polynomial time algorithm for finding Ω(OPT^1/p) edge-disjoint paths for some p>1, then, for some α >0, we can give a randomized O(n^1/2-α)-approximation algorithm for the maximum edge-disjoint paths problem by using Rao-Zhou's algorithm. 2. Based on the well-linked set of Chekuri, Khanna, and Shepherd, we show that there is a randomized algorithm for finding Ω(OPT^1/4) edge-disjoint paths connecting given terminal pairs with congestion two.

Our framework for this algorithm is more general. Indeed, the above two ingredients also work for the maximum edge-disjoint paths problem (with congestion one) if the following conjecture is true. Conjecture: There is a (randomized) polynomial time algorithm for finding Ω(OPT^1/p/β(n)) edge-disjoint paths connecting given terminal pairs, where β is a poly-logarithmic function.

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References

[1]

. Andrews,Approximation algorithms for the edge-disjoint paths problem via Racke decompositions, Proc. 51st IEEE Symposium on Foundations of Computer Science (FOCS), 2010, 277--286.

Digital Library

[2]

. Andrews, J. Chuzhoy, S. Khanna and L. Zhang,Hardness of the undirected edge-disjoint paths problem with congestion. Proc. 46th IEEE Symposium on Foundations of Computer Science (FOCS), 2005,226--244.

Digital Library

[3]

C. Chekuri, S. Khanna and B. Shepherd,The all-or-nothing multicommodity flow problem, Proc. 36th ACM Symposium on Theory of Computing (STOC), 2004, 156--165.

Digital Library

[4]

C. Chekuri, S. Khanna and B. Shepherd,Multicommodity flow, well-linked terminals, and routing problems, Proc. 37th ACM Symposium on Theory of Computing (STOC), 2005, 183--192.

Digital Library

[5]

C. Chekuri, S. Khanna and B. Shepherd,An O(√n) approximation and integrality gap for disjoint paths and unsplittable flow, Theory of computing, 2 (2006), 137--146.

[6]

. Diestel, Graph Theory, 3rd Edition, Springer-Verlag, 2005.

[7]

. Diestel, T.R. Jensen, K.Y. Gorbunov and C. Thomassen,Highly connected sets and the excluded grid theorem, J. Combin. Theory Ser. B, 75 (1999), 61--73.

Digital Library

[8]

S. Even, A. Itai and A. Shamir, On the complexity of timetable and multicommodity flow problems, SIAM Journal on Computing, 5 (1976), 691--703.

[9]

. Frank,Packing paths, cuts and circuits -- a survey,in Paths, Flows and VLSI-Layout,B. Korte, L. Lovász, H. J. Promel and A. Schrijver (Eds.),Springer-Verlag, Berlin, 1990, 49--100.

[10]

. Guruswami, S. Khanna, R. Rajaraman, B. Shepherd, and M. Yannakakis,Near-optimal hardness results and approximaiton algorithms for edge-disjoint paths andrelated problems, J. Comp. Syst. Science, 67 (2003), 473--496.Also, Proc. 31st ACM Symposium on Theory of Computing (STOC), 1999, 19--28.

Digital Library

[11]

. Kawarabayashi and B. Reed,A nearly linear time algorithm for the half integral disjoint paths packing, Proc. ACM-SIAM Symposium on Discrete Algorithms (SODA), 2008, 446--454.

Digital Library

[12]

. Kawarabayashi and Y. Kobayashi, Improved algorithm for the half-disjoint paths problem, Proc. 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2010, 287--297.

Digital Library

[13]

. Kawarabayashi and Y. Kobayashi, An O(log n)-approximation algorithm for the disjoint paths problem in Eulerian planar graphs and 4-edge-connected planar graphs, Proc. 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2010, 274--286.

Digital Library

[14]

. Kawarabayashi, Y. Kobayashi and B. Reed,The disjoint paths problem in quadratic time, submitted.Available at http://research.nii.ac.jp/k_keniti/quaddp1.pdf.

[15]

. Kleinberg,Decision algorithms for unsplittable flow and the half-disjointpaths problem, Proc. 30th ACM Symposium on Theory of Computing (STOC), 1998,530--539.

Digital Library

[16]

. Kleinberg,An approximation algorithm for the disjoint paths problem in even-degree planar graphs, Proc. 46th IEEE Symposium on Foundations of Computer Science (FOCS), 2005, 627--636.

Digital Library

[17]

. Kleinberg and É. Tardos,Disjoint paths in densely embedded graphs, Proc. 36th IEEE Symposium on Foundations of Computer Science (FOCS), 1995, 52--61.

Digital Library

[18]

. Kleinberg and É. Tardos,Approximations for the disjoint paths problem in high-diameter planar networks, Proc. 27th ACM Symposium on Theory of Computing (STOC), 1995, 26--35.

Digital Library

[19]

. Kolliopoulos and C. Stein,Improved approximation algorithms for unsplittable flow problems, Proc. 38th IEEE Symposium on Foundations of Computer Science (FOCS), 1997, 426--435.

Digital Library

[20]

M.R. Kramer and J. van Leeuwen,The complexity of wire-routing and finding minimum area layouts for arbitrary VLSI circuits, Adv. Comput. Res., 2 (1984), 129--146.

[21]

. Kreutzer and S. Tazari,On brambles, grid-like minors, and parameterized intractability of monadicsecond-order logic, Proc. ACM-SIAM Symposium on Discrete Algorithms (SODA), 2010, 354--364.

Digital Library

[22]

. Raghavan and C.D. Thompson,Randomized rounding: A technique for provably good algorithms and algorithmic proofs, Combinatorica, 7 (1987), 365--374.

Digital Library

[23]

. Rao and S. Zhou,Edge disjoint paths in moderately connected graphs, SIAM J. Computing, 39 (2010), 1856--1887.

Digital Library

[24]

. Reed,Tree width and tangles: A new connectivity measure and some applications,in Surveys in Combinatorics,London Math. Soc. Lecture Note Ser. 241,Cambridge Univ. Press, Cambridge, 1997, 87--162.

[25]

. Reed and D. Wood,Polynomial treewidth forces a large grid-like minor, availableat arXiv:0809.0724v3{math.CO}, 2008.

[26]

. Robertson and P.D. Seymour,Graph minors. XIII. The disjoint paths problem, J. Combin. Theory Ser. B, 63 (1995), 65--110.

Digital Library

[27]

. Robertson, P.D. Seymour and R. Thomas,Quickly excluding a planar graph, J. Combin. Theory Ser. B, 62 (1994), 323--348.

Digital Library

[28]

. Schrijver, Combinatorial Optimization: Polyhedra and Efficiency, number 24 in Algorithm and Combinatorics, Springer-Verlag, 2003.

[29]

. Srinviasan,Improved approximations for edge-disjoint paths, unsplittable flow, andrelated routing problems, Proc. 38th IEEE Symposium on Foundations of Computer Science (FOCS), 1997, 416--425.

Digital Library

Cited By

Kawarabayashi KKobayashi Y(2016)An Improved Approximation Algorithm for the Edge-Disjoint Paths Problem with Congestion TwoACM Transactions on Algorithms10.1145/296041013:1(1-17)Online publication date: 21-Sep-2016
https://dl.acm.org/doi/10.1145/2960410
Chuzhoy JLi S(2016)A Polylogarithmic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2Journal of the ACM10.1145/289347263:5(1-51)Online publication date: 8-Nov-2016
https://dl.acm.org/doi/10.1145/2893472
Chekuri CEne AKhanna S(2013)Poly-logarithmic approximation for maximum node disjoint paths with constant congestionProceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms10.5555/2627817.2627841(326-341)Online publication date: 6-Jan-2013
https://dl.acm.org/doi/10.5555/2627817.2627841
Show More Cited By

Index Terms

Breaking o(n^1/2)-approximation algorithms for the edge-disjoint paths problem with congestion two
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms
      2. Paths and connectivity problems

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cover image ACM Conferences

STOC '11: Proceedings of the forty-third annual ACM symposium on Theory of computing

June 2011

840 pages

ISBN:9781450306911

DOI:10.1145/1993636

General Chair:
Lance Fortnow
Northwestern University
,
Program Chair:
Salil Vadhan
Harvard University

Copyright © 2011 ACM.

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Published: 06 June 2011

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STOC'11: Symposium on Theory of Computing

June 6 - 8, 2011

California, San Jose, USA

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STOC '11 Paper Acceptance Rate 84 of 304 submissions, 28%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Cited By

Kawarabayashi KKobayashi Y(2016)An Improved Approximation Algorithm for the Edge-Disjoint Paths Problem with Congestion TwoACM Transactions on Algorithms10.1145/296041013:1(1-17)Online publication date: 21-Sep-2016
https://dl.acm.org/doi/10.1145/2960410
Chuzhoy JLi S(2016)A Polylogarithmic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2Journal of the ACM10.1145/289347263:5(1-51)Online publication date: 8-Nov-2016
https://dl.acm.org/doi/10.1145/2893472
Chekuri CEne AKhanna S(2013)Poly-logarithmic approximation for maximum node disjoint paths with constant congestionProceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms10.5555/2627817.2627841(326-341)Online publication date: 6-Jan-2013
https://dl.acm.org/doi/10.5555/2627817.2627841
Chuzhoy JKarloff HPitassi T(2012)Routing in undirected graphs with constant congestionProceedings of the forty-fourth annual ACM symposium on Theory of computing10.1145/2213977.2214054(855-874)Online publication date: 19-May-2012
https://dl.acm.org/doi/10.1145/2213977.2214054
Chuzhoy JLi S(2012)A Polylogarithmic Approximation Algorithm for Edge-Disjoint Paths with Congestion 2Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science10.1109/FOCS.2012.54(233-242)Online publication date: 20-Oct-2012
https://dl.acm.org/doi/10.1109/FOCS.2012.54

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