TY - GEN

T1 - Approximation algorithms and hardness of the k-route cut problem

AU - Chuzhoy, Julia

AU - Makarychev, Yury

AU - Vijayaraghavan, Aravindan

AU - Zhou, Yuan

PY - 2012

Y1 - 2012

N2 - We study the k-route cut problem: given an undirected edge-weighted graph G = (V, E), a collection {(s1, t1), (s2, t 2), . . . , (sr, tr)} of source-sink pairs, and an integer connectivity requirement k, the goal is to find a minimum-weight subset E′ of edges to remove, such that the connectivity of every pair (si, ti) falls below k. Specifically, in the edge-connectivity version, EC-kRC, the requirement is that there are at most (k - 1) edge-disjoint paths connecting si to ti in G\E′, while in the vertex-connectivity version, VC-kRC, the same requirement is for vertex-disjoint paths. Prior to our work, poly-logarithmic approximation algorithms have been known for the special case where k ≤ 3, but no non-trivial approximation algorithms were known for any value k > 3, except in the single-source setting. We show an O(k log3/2 r)-approximation algorithm for EC-kRC with uniform edge weights, and several polylogarithmic bi-criteria approximation algorithms for EC-kRC and VC-kRC, where the connectivity requirement k is violated by a constant factor. We complement these upper bounds by proving that VC-kRC is hard to approximate to within a factor of kε for some fixed ε > 0. We then turn to study a simpler version of VC-kRC, where only one source-sink pair is present. We give a simple bi-criteria approximation algorithm for this case, and show evidence that even this restricted version of the problem may be hard to approximate. For example, we prove that the single source-sink pair version of VC-kRC has no constant-factor approximation, assuming Feige's Random κ-AND assumption.

AB - We study the k-route cut problem: given an undirected edge-weighted graph G = (V, E), a collection {(s1, t1), (s2, t 2), . . . , (sr, tr)} of source-sink pairs, and an integer connectivity requirement k, the goal is to find a minimum-weight subset E′ of edges to remove, such that the connectivity of every pair (si, ti) falls below k. Specifically, in the edge-connectivity version, EC-kRC, the requirement is that there are at most (k - 1) edge-disjoint paths connecting si to ti in G\E′, while in the vertex-connectivity version, VC-kRC, the same requirement is for vertex-disjoint paths. Prior to our work, poly-logarithmic approximation algorithms have been known for the special case where k ≤ 3, but no non-trivial approximation algorithms were known for any value k > 3, except in the single-source setting. We show an O(k log3/2 r)-approximation algorithm for EC-kRC with uniform edge weights, and several polylogarithmic bi-criteria approximation algorithms for EC-kRC and VC-kRC, where the connectivity requirement k is violated by a constant factor. We complement these upper bounds by proving that VC-kRC is hard to approximate to within a factor of kε for some fixed ε > 0. We then turn to study a simpler version of VC-kRC, where only one source-sink pair is present. We give a simple bi-criteria approximation algorithm for this case, and show evidence that even this restricted version of the problem may be hard to approximate. For example, we prove that the single source-sink pair version of VC-kRC has no constant-factor approximation, assuming Feige's Random κ-AND assumption.

UR - http://www.scopus.com/inward/record.url?scp=84860189048&partnerID=8YFLogxK

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U2 - 10.1137/1.9781611973099.63

DO - 10.1137/1.9781611973099.63

M3 - Conference contribution

AN - SCOPUS:84860189048

SN - 9781611972108

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 780

EP - 799

BT - Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012

PB - Association for Computing Machinery

T2 - 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012

Y2 - 17 January 2012 through 19 January 2012

ER -