In this paper we study approximation algorithms for multi-route cut problems in undirected graphs. In these problems the goal is to find a minimum cost set of edges to be removed from a given graph such that the edge-connectivity (or node-connectivity) between certain pairs of nodes is reduced below a given threshold K. In the usual cut problems the edge connectivity is required to be reduced below 1 (i.e. disconnected). We consider the case of K = 2 and obtain poly-logarithmic approximation algorithms for fundamental cut problems including single-source, multiway-cut, multicut, and sparsest cut. These cut problems are dual to multi-route flows that are of interest in fault-tolerant networks flows. Our results show that the flow-cut gap between 2-route cuts and 2-route flows is poly-logarithmic in undirected graphs with arbitrary capacities. 2-route cuts are also closely related to well-studied feedback problems and we obtain results on some new variants. Multi-route cuts pose interesting algorithmic challenges. The new techniques developed here are of independent technical interest, and may have applications to other cut and partitioning problems.

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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 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 ECkRC 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 kɛ for some fixed ɛ> 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 VCkRC has no constant-factor approximation, assuming Feige’s Random κ-AND assumption.

Given an integer h, a graph G = (V, E) with arbitrary positive edge capacities and k pairs of vertices (s1, t1), (s2, t2),..., (sk, tk), called terminals, an h-route cut is a set F ⊆ E of edges such that after the removal of the edges in F no pair si−ti is connected by h edge-disjoint paths (i.e., the connectivity of every si − ti pair is at most h − 1 in (V, E\F)). The h-route cut is a natural generalization of the classical cut problem for multicommodity flows (take h = 1). The main result of this paper is an O(h 5 2 2h (h + log k) 2)-approximation algorithm for the minimum h-route cut problem in the case that s1 = s2 = · · · = sk, called the single source case. As a corollary of it we obtain an approximate duality theorem for multiroute multicommodity flows and cuts with a single source. This partially answers an open question posted in several previous papers dealing with cuts for multicommodity multiroute problems. 1

Abstract An elementary h-route flow, for an integer h ≥ 1, is a set of h edge-disjoint paths between a source and a sink, each path carrying a unit of flow, and an h-route flow is a non-negative linear combination of elementary h-route flows. An h-route cut is a set of edges whose removal decreases the maximum h-route flow between a given source-sink pair (or between every source-sink pair in the multicommodity setting) to zero. The main result of this paper is an approximate duality theorem for multicommodity h-route cuts and flows, for h ≤ 3: The size of a minimum h-route cut is at least f/h and at most O(log 4 k · f) where f is the size of the maximum h-route flow and k is the number of commodities. The main step towards the proof of this duality is the design and analysis of a polynomial-time approximation algorithm for the minimum h-route cut problem for h = 3 that has an approximation ratio of O(log 4 k). Previously, polylogarithmic approximation was known only for h-route cuts for h ≤ 2. A key ingredient of our algorithm is a novel rounding technique that we call multilevel ball-growing. Though the proof of the duality relies on this algorithm, it is not a straightforward corollary of it as in the case of classical multicommodity flows and cuts. Similar results are shown also for the sparsest multiroute cut problem.