Approximation Algorithms and Hardness of thek-Route Cut Problem [chapter]

Julia Chuzhoy, Yury Makarychev, Aravindan Vijayaraghavan, Yuan Zhou
2012 Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms  
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
more » ... 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 algorithm has been known for the special case where k = 2, but no non-trivial approximation algorithms were known for any value k > 2, except in the single-source setting. We show an O(k log 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 present 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. . Work done while visiting Toyota Technological Institute, Chicago that for each 1 ≤ i ≤ r, the number of edge-disjoint paths connecting s i to t i in graph G \ E is less than k. In the vertex-connectivity version (VC-kRC), the requirement is that the number of vertex-disjoint paths connecting s i to t i is less than k. It is not hard to see that VC-kRC captures EC-kRC as a special case (see the full version ), and hence is more general. It is also easy to see that minimum multicut is a special case of both EC-kRC and VC-kRC, with the connectivity requirement k = 1. We also consider a special case of EC-kRC, where all edges have unit weight, and we refer to it as the uniform EC-kRC. We note that for VC-kRC, the uniform and the non-uniform edge-weight versions are equivalent up to a small loss in the approximation factor, as shown in the full version of the paper, and so we do not distinguish between them. The primary motivation for studying k-route cuts comes from multi-commodity flows in fault tolerant settings, where the resilience to edge and node failures is important. An elementary k-route flow between a pair s and t of vertices is a set of k disjoint paths connecting s to t. A k-route (st)-flow is just a combination of such elementary k-route flows, where each elementary flow is assigned some fractional value. This is a natural generalization of the standard (st)-flows, which ensures that the flow is resilient to the failure of up to (k − 1) edges or vertices. Multi-route flows were first introduced by Kishimoto [Kis96], and have since been studied in the context of communication networks [BCSK07, BCK03, ACKN07]. In a series of papers, Kishimoto [Kis96], Kishimoto and Takeuchi [KT93] and Aggarwal and Orlin [AO02] have developed a number of efficient algorithms for computing maximum multi-route flows. As in the case of standard flows, we can extend kroute (st)-flows to the multi-commodity setting, where the goal is to maximize the total k-route flow among all source-destination pairs. It is easy to see that the minimum k-route cut is a natural upper bound on the maximum k-route flow -just like minimum multicut upper-bounds the value of the maximum multicommodity flow. Hence, as in the case with the standard multicut, multi-route cuts can be seen as revealing the network bottleneck, and so the minimum k-route cut in a graph captures the robustness of real-life computer and transportation networks. The first approximation algorithm for the EC-kRC problem, due to Chekuri and Khanna [CK08], achieved a factor O(log 2 n log r)-approximation for the special case where k = 2, by rounding a Linear Programming relaxation. This was improved by Barman and Chawla [BC10] to give an O(log 2 r)-approximation algorithm for the same version, by generalizing the regiongrowing LP-rounding scheme of [LR99, GVY95]. They
doi:10.1137/1.9781611973099.63 dblp:conf/soda/ChuzhoyMVZ12 fatcat:2g3vjlapfbga3n63jdhlc23dte