Region growing for multi-route cuts

Abstract

We study a number of multi-route cut problems: given a graph G = (V, E) and connectivity thresholds k (u,v) on pairs of nodes, the goal is to find a minimum cost set of edges or vertices the removal of which reduces the connectivity between every pair (u, v) to strictly below its given threshold. These problems arise in the context of reliability in communication networks; They are natural generalizations of traditional minimum cut problems where the thresholds are either 1 (we want to completely separate the pair) or ∞ (we don't care about the connectivity for the pair). We provide the first non-trivial approximations to a number of variants of the problem including for both node-disjoint and edge-disjoint connectivity thresholds. A main contribution of our work is an extension of the region growing technique for approximating minimum multicuts to the multi-route setting. When the connectivity thresholds are either 2 or ∞ (the "2-route cut" case), we obtain polylogarithmic approximations while satisfying the thresholds exactly. For arbitrary connectivity thresholds this approach leads to bicriteria approximations where we approximately satisfy the thresholds and approximately minimize the cost. We present a number of different algorithms achieving different cost-connectivity tradeoffs.

Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing

“…The approximation ratio of our algorithm is O(log 4 k). This provides a partial answer to open problems of several papers (Bruhn et al [6], Chekuri and Khanna [7] and Barman and Chawla [5]). The 3-route cut problem is more complicated than the 1-route and 2-route cut problems: while 1-route and 2-route cuts separate the graph into independent parts, h-route cuts do not have this property for h > 2.…”

“…This is the key observation of Barman and Chawla [5] (proved in a different way). We add the edges from δ 1 (r) to the 2-route cut that we construct, remove the ball B(r) from the graph (observe that after the removal of δ 1 (r), no terminal t j in B(r) is 2-connected with s) and proceed with the next iteration.…”

“…As their algorithm is based on an LP relaxation that is dual to the LP for the maximum 2-route flow problem, an implicit corollary of their result is an approximate duality of 2-route flows and 2-route cuts. The approximation factor for 2-route cuts was recently improved by Barman and Chawla [5] who described an O(log 2 k)-approximation for the 2-route cut problem. Their algorithm is based on a different linear programming relaxation that allows them to extend the classical (discrete) ball-growing (or region-growing) technique (cf.…”

“…Though the proof of the duality relies on the approximation algorithm, it is not a straightforward corollary of it as is the case for classical multicommodity flows and cuts. For the duality proof we show a tight relationship between two different linear programming relaxations [5,7] of the h-route cut problem; our main tool here is the Farkas' lemma.…”

“…[17,5,18,19]): while there exists a vertex t i that is connected by three edge disjoint paths to s, perform a 3-route cut separating t i and s, charge the 3-route cut to a certain part (volume) of the network (roughly, to the region that was used to define the cut) and proceed with the next iteration. The 3-route cut in each iteration is derived from the fractional solution of the LP relaxation of the minimum 3-route cut problem.…”