### Abstract

We study the k-route cut problem: given an undirected edge-weighted graph G = (V, E), a collection {(s_{1}, t_{1}), (s_{2}, t_{2}), ⋯, (s_{r}, t_{r})} 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 (s_{i}, t_{i}) 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 s_{i} to t_{i} 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(klog^{3/2} 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 k^{∈} for some fixed ∈ > 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.

Original language | English (US) |
---|---|

Article number | 2 |

Journal | ACM Transactions on Algorithms |

Volume | 12 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2015 |

Externally published | Yes |

### Fingerprint

### Keywords

- Approximation algorithm
- K-route cut problem

### ASJC Scopus subject areas

- Mathematics (miscellaneous)

### Cite this

*ACM Transactions on Algorithms*,

*12*(1), [2]. https://doi.org/10.1145/2644814

**Approximation algorithms and hardness of the k-route cut problem.** / Chuzhoy, Julia; Makarychev, Yury; Vijayaraghavan, Aravindan; Zhou, Yuan.

Research output: Contribution to journal › Article

*ACM Transactions on Algorithms*, vol. 12, no. 1, 2. https://doi.org/10.1145/2644814

}

TY - JOUR

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

AU - Chuzhoy, Julia

AU - Makarychev, Yury

AU - Vijayaraghavan, Aravindan

AU - Zhou, Yuan

PY - 2015/12/1

Y1 - 2015/12/1

N2 - We study the k-route cut problem: given an undirected edge-weighted graph G = (V, E), a collection {(s1, t1), (s2, t2), ⋯, (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(klog3/2 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 k∈ for some fixed ∈ > 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, t2), ⋯, (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(klog3/2 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 k∈ for some fixed ∈ > 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.

KW - Approximation algorithm

KW - K-route cut problem

UR - http://www.scopus.com/inward/record.url?scp=84954200150&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84954200150&partnerID=8YFLogxK

U2 - 10.1145/2644814

DO - 10.1145/2644814

M3 - Article

AN - SCOPUS:84954200150

VL - 12

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

SN - 1549-6325

IS - 1

M1 - 2

ER -