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12
An O(k 3 log n)Approximation Algorithm for VertexConnectivity Survivable Network Design
, 2008
"... In the Survivable Network Design problem (SNDP), we are given an undirected graph G(V, E) with costs on edges, along with a connectivity requirement r(u, v) for each pair u, v of vertices. The goal is to find a minimumcost subset E ∗ of edges, that satisfies the given set of pairwise connectivity r ..."
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Cited by 24 (0 self)
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In the Survivable Network Design problem (SNDP), we are given an undirected graph G(V, E) with costs on edges, along with a connectivity requirement r(u, v) for each pair u, v of vertices. The goal is to find a minimumcost subset E ∗ of edges, that satisfies the given set of pairwise connectivity requirements. In the edgeconnectivity version we need to ensure that there are r(u, v) edgedisjoint paths for every pair u, v of vertices, while in the vertexconnectivity version the paths are required to be vertexdisjoint. The edgeconnectivity version of SNDP is known to have a 2approximation. However, no nontrivial approximation algorithm has been known so far for the vertex version of SNDP, except for special cases of the problem. We present an extremely simple algorithm to achieve an O(k 3 log n)approximation for this problem, where k denotes the maximum connectivity requirement, and n denotes the number of vertices. We also give a simple proof of the recently discovered O(k 2 log n)approximation result for the singlesource version of vertexconnectivity SNDP. We note that in both cases, our analysis in fact yields slightly better guarantees in that the log n term in the approximation guarantee can be replaced with a log τ term where τ denotes the number of distinct vertices that participate in one or more pairs with a positive connectivity requirement.
Algorithms for SingleSource Vertex Connectivity
"... In the Survivable Network Design Problem (SNDP) the goal is to find a minimum cost subset of edges that satisfies a given set of pairwise connectivity requirements among the vertices. This general network design framework has been studied extensively and is tied to the development of major algorithm ..."
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Cited by 22 (2 self)
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In the Survivable Network Design Problem (SNDP) the goal is to find a minimum cost subset of edges that satisfies a given set of pairwise connectivity requirements among the vertices. This general network design framework has been studied extensively and is tied to the development of major algorithmic techniques. For the edgeconnectivity version of the problem, a 2approximation algorithm is known for arbitrary pairwise connectivity requirements. However, no nontrivial algorithms are known for its vertex connectivity counterpart. In fact, even highly restricted special cases of the vertex connectivity version remain poorly understood. We study the singlesource kvertex connectivity version of SNDP. We are given a graph G(V, E) with a subset T of terminals and a source vertex s, and the goal is to find a minimum cost subset of edges ensuring that every terminal is kvertex connected to s. Our main result is an O(k log n)approximation algorithm for this problem; this improves upon the recent 2 O(k2) log 4 napproximation. Our algorithm is based on an intuitive rerouting scheme. The analysis relies on a structural result that may be of independent interest: we show that any solution can be decomposed into a disjoint collection of multiplelegged spiders, which are then used to reroute flow from terminals to the source via other terminals. We also obtain the first nontrivial approximation algorithm for the vertexcost version of the same problem, achieving an O(k 7 log 2 n)approximation. 1.
Online and Stochastic Survivable Network Design
"... Consider the edgeconnectivity survivable network design problem: given a graph G = (V, E) with edgecosts, and edgeconnectivity requirements rij ∈ Z≥0 for every pair of vertices i, j ∈ V, find an (approximately) minimumcost network that provides the required connectivity. While this problem is kno ..."
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Cited by 10 (2 self)
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Consider the edgeconnectivity survivable network design problem: given a graph G = (V, E) with edgecosts, and edgeconnectivity requirements rij ∈ Z≥0 for every pair of vertices i, j ∈ V, find an (approximately) minimumcost network that provides the required connectivity. While this problem is known to admit good approximation algorithms in the offline case, no algorithms were known for this problem in the online setting. In this paper, we give a randomized O(rmax log 3 n) competitive online algorithm for this edgeconnectivity network design problem, where rmax = maxij rij. Our algorithms use the standard embeddings of graphs into random subtrees (i.e., into singly connected subgraphs) as an intermediate step to get algorithms for higher connectivity. Our results for the online problem give us approximation algorithms that admit strict costshares with the same strictness value. This, in turn, implies approximation algorithms for (a) the rentorbuy version and (b) the (twostage) stochastic version of the edgeconnected network design problem with independent arrivals. For these two problems, if we are in the case when the underlying graph is complete and the edgecosts are metric (i.e., satisfy the triangle inequality), we improve our results to give O(1)strict cost shares, which gives constantfactor rentorbuy and stochastic algorithms for these instances.
A note on rooted survivable networks
 IPL
, 2009
"... The (undirected) Rooted Survivable Network Design (Rooted SND) problem is: given a complete graph on node set V with edgecosts, a root s ∈ V, and (node)connectivity requirements {r(t) : t ∈ T ⊆ V}, find a minimum cost subgraph G that contains r(t) internallydisjoint stpaths for all t ∈ T. For la ..."
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Cited by 8 (2 self)
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The (undirected) Rooted Survivable Network Design (Rooted SND) problem is: given a complete graph on node set V with edgecosts, a root s ∈ V, and (node)connectivity requirements {r(t) : t ∈ T ⊆ V}, find a minimum cost subgraph G that contains r(t) internallydisjoint stpaths for all t ∈ T. For large values of k = maxt∈T r(t) Rooted SND is at least as hard to approximate as Directed Steiner Tree [Lando & Nutov, APPROX 2008]. For Rooted SND [Chuzhoy & Khanna, FOCS 08] gave recently an approximation algorithm with ratio O(k 2 log n). Independently, and using different techniques, we obtained at the same time a simpler primaldual algorithm with the same ratio. 1
A graph reduction step preserving elementconnectivity and applications
 in International Colloquium on Automata, Languages and Programming
"... Given an undirected graph G = (V, E) and subset of terminals T ⊆ V, the elementconnectivity κ ′ G (u, v) of two terminals u, v ∈ T is the maximum number of uv paths that are pairwise disjoint in both edges and nonterminals V \ T (the paths need not be disjoint in terminals). Elementconnectivity ..."
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Cited by 5 (1 self)
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Given an undirected graph G = (V, E) and subset of terminals T ⊆ V, the elementconnectivity κ ′ G (u, v) of two terminals u, v ∈ T is the maximum number of uv paths that are pairwise disjoint in both edges and nonterminals V \ T (the paths need not be disjoint in terminals). Elementconnectivity is more general than edgeconnectivity and less general than vertexconnectivity. Hind and Oellermann [21] gave a graph reduction step that preserves the global elementconnectivity of the graph. We show that this step also preserves local connectivity, that is, all the pairwise elementconnectivities of the terminals. We give two applications of this reduction step to connectivity and network design problems. • Given a graph G and disjoint terminal sets T1, T2,..., Tm, we seek a maximum number of elementdisjoint Steiner forests where each forest connects each Ti. We prove that if each Ti is k element k connected then there exist Ω( log hlog m) elementdisjoint Steiner forests, where h =  i Ti. If G is planar (or more generally, has fixed genus), we show that there exist Ω(k) Steiner forests. Our proofs are constructive, giving polytime algorithms to find these forests; these are the first nontrivial algorithms for packing elementdisjoint Steiner Forests. • We give a very short and intuitive proof of a spiderdecomposition theorem of Chuzhoy and Khanna [12] in the context of the singlesink kvertexconnectivity problem; this yields a simple and alternative analysis of an O(k log n) approximation. Our results highlight the effectiveness of the elementconnectivity reduction step; we believe it will find more applications in the future. 1
Tree Embeddings for TwoEdgeConnected Network Design
"... The group Steiner problem is a classical network design problem where we are given a graph and a collection of groups of vertices, and want to build a mincost subgraph that connects the root vertex to at least one vertex from each group. What if we wanted to build a subgraph that twoedgeconnects ..."
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The group Steiner problem is a classical network design problem where we are given a graph and a collection of groups of vertices, and want to build a mincost subgraph that connects the root vertex to at least one vertex from each group. What if we wanted to build a subgraph that twoedgeconnects the root to each group—that is, for every group g ⊆ V, the subgraph should contain two edgedisjoint paths from the root to some vertex in g? What if we wanted the two edgedisjoint paths to end up at distinct vertices in the group, so that the loss of a single member of the group would not destroy connectivity? In this paper, we investigate treeembedding techniques that can be used to solve these and other 2edgeconnected network design problems. We illustrate the potential of these techniques by giving polylogarithmic approximation algorithms for twoedgeconnected versions of the group Steiner, connected facility location, buyatbulk, and the kMST problems. 1
Approximation Algorithms for Network Design: A Survey
"... In a typical instance of a network design problem, we are given a directed or undirected graph G = (V,E), nonnegative edgecosts ce for all e ∈ E, and our goal is to find a minimumcost subgraph H of G that satisfies some design criteria. For example, we may wish to find a minimumcost set of edges ..."
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Cited by 3 (0 self)
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In a typical instance of a network design problem, we are given a directed or undirected graph G = (V,E), nonnegative edgecosts ce for all e ∈ E, and our goal is to find a minimumcost subgraph H of G that satisfies some design criteria. For example, we may wish to find a minimumcost set of edges that induces a connected graph (this is the minimumcost spanning tree problem), or we might want to find a minimumcost set of arcs in a directed graph such that every vertex can reach every other vertex (this is the minimumcost strongly connected subgraph problem). This abstract model for network design problems has a large number of practical applications; the design process of telecommunication and traffic networks, and VLSI chip design are just two examples. Many practically relevant instances of network design problems are NPhard, and thus likely intractable. This survey focuses on approximation algorithms as one possible way of circumventing this impasse. Approximation algorithms are efficient (i.e., they run in polynomialtime), and they compute solutions to a given instance of an optimization problem whose objective values are close to those of the respective optimum solutions. More concretely, most of the problems discussed in this survey are minimization problems. We then say that an algorithm is an αapproximation for a given problem if the ratio of the cost of an approximate solution computed by the algorithm to that of an optimum solution is at most α over all instances. In the
Approximating FaultTolerant GroupSteiner Problems
 LEIBNIZ INTERNATIONAL PROCEEDINGS IN INFORMATICS
, 2009
"... In this paper, we initiate the study of designing approximation algorithms for FaultTolerant GroupSteiner (FTGS) problems. The motivation is to protect the wellstudied groupSteiner networks from edge or vertex failures. In FaultTolerant GroupSteiner problems, we are given a graph with edge (o ..."
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In this paper, we initiate the study of designing approximation algorithms for FaultTolerant GroupSteiner (FTGS) problems. The motivation is to protect the wellstudied groupSteiner networks from edge or vertex failures. In FaultTolerant GroupSteiner problems, we are given a graph with edge (or vertex) costs, a root vertex, and a collection of subsets of vertices called groups. The objective is to find a minimumcost subgraph that has two edge (or vertex) disjoint paths from each group to the root. We present approximation algorithms and hardness results for several variants of this basic problem, e.g., edgecosts vs. vertexcosts, edgeconnectivity vs. vertexconnectivity, and 2connecting from each group a single vertex vs. many vertices. Main contributions of our paper include the introduction of very general structural lemmas on connectivity and a charging scheme that may find more applications in the future. Our algorithmic results are supplemented by inapproximability results, which are tight in some cases. Our algorithms employ a variety of techniques. For the edgeconnectivity variant, we use a primaldual based algorithm for covering an uncrossable setfamily, while for the vertexconnectivity version, we prove a new graphtheoretic lemma that shows equivalence between obtaining two vertexdisjoint paths from two vertices and 2connecting a carefully chosen single vertex. To handle large groupsizes, we use a pSteiner tree algorithm to identify the “correct” pair of terminals from each group to be connected to the root. We also use a nontrivial charging scheme to improve the approximation ratio for the most general problem we consider.
Approximation Algorithms for NETWORK DESIGN AND ORIENTEERING
, 2010
"... This thesis presents approximation algorithms for some N PHard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widelybelieved complexitytheoretic assumption that P ̸ = N P, there are no efficient (i.e., polynomialt ..."
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This thesis presents approximation algorithms for some N PHard combinatorial optimization problems on graphs and networks; in particular, we study problems related to Network Design. Under the widelybelieved complexitytheoretic assumption that P ̸ = N P, there are no efficient (i.e., polynomialtime) algorithms that solve these problems exactly. Hence, if one desires efficient algorithms for such problems, it is necessary to consider approximate solutions: An approximation algorithm for an N PHard problem is a polynomial time algorithm which, for any instance of the problem, finds a solution whose value is guaranteed to be within a multiplicative factor ρ of the value of an optimal solution to that instance. We attempt to design algorithms for which this factor ρ, referred to as the approximation ratio of the algorithm, is as small as possible. The field of Network Design comprises a large class of problems that deal with constructing networks of low cost and/or high capacity, routing data through existing networks, and many related issues. In this thesis, we focus chiefly on designing faulttolerant networks. Two vertices u, v in a network are said to be kedgeconnected if deleting any set of k − 1 edges leaves u and v connected; similarly, they are kvertex connected if deleting any set of k − 1 other vertices or edges leaves u and v connected. We focus on building networks that are highly connected, meaning