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266
Approximation Algorithms for Directed Steiner Problems
 Journal of Algorithms
, 1998
"... We give the first nontrivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ratios known before our work we ..."
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Cited by 178 (8 self)
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We give the first nontrivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ratios known before our work were the trivial O(k)approximations. For the directed Steiner tree problem, we design a family of algorithms that achieves an approximation ratio of i(i \Gamma 1)k 1=i in time O(n i k 2i ) for any fixed i ? 1, where k is the number of terminals. Thus, an O(k ffl ) approximation ratio can be achieved in polynomial time for any fixed ffl ? 0. Setting i = log k, we obtain an O(log 2 k) approximation ratio in quasipolynomial time. For the directed generalized Steiner network problem, we give an algorithm that achieves an approximation ratio of O(k 2=3 log 1=3 k), where k is the number of pairs of vertices that are to be connected. Related problems including the group Steiner...
The primaldual method for approximation algorithms and its application to network design problems.
, 1997
"... Abstract In this survey, we give an overview of a technique used to design and analyze algorithms that provide approximate solutions to N P hard problems in combinatorial optimization. Because of parallels with the primaldual method commonly used in combinatorial optimization, we call it the prim ..."
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Cited by 137 (5 self)
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Abstract In this survey, we give an overview of a technique used to design and analyze algorithms that provide approximate solutions to N P hard problems in combinatorial optimization. Because of parallels with the primaldual method commonly used in combinatorial optimization, we call it the primaldual method for approximation algorithms. We show how this technique can be used to derive approximation algorithms for a number of different problems, including network design problems, feedback vertex set problems, and facility location problems.
Provisioning a Virtual Private Network: A network design problem for multicommodity flow
 In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing
, 2001
"... Consider a setting in which a group of nodes, situated in a large underlying network, wishes to reserve bandwidth on which to support communication. Virtual private networks (VPNs) are services that support such a construct; rather than building a new physical network on the group of nodes that must ..."
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Cited by 109 (13 self)
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Consider a setting in which a group of nodes, situated in a large underlying network, wishes to reserve bandwidth on which to support communication. Virtual private networks (VPNs) are services that support such a construct; rather than building a new physical network on the group of nodes that must be connected, bandwidth in the underlying network is reserved for communication within the group, forming a virtual “subnetwork.” Provisioning a virtual private network over a set of terminals gives rise to the following general network design problem. We have bounds on the cumulative amount of traffic each terminal can send and receive; we must choose a path for each pair of terminals, and a bandwidth allocation for each edge of the network, so that any traffic matrix consistent with the given upper bounds can be feasibly routed. Thus, we are seeking to design a network that can support a continuum of possible traffic scenarios. We provide optimal and approximate algorithms for several variants of this problem, depending on whether the traffic matrix is required to be symmetric, and on whether the designed network is required to be a tree (a natural constraint in a number of basic applications). We also establish a relation between this collection of network design problems and a variant of the facility location problem introduced by Karger and Minkoff; we extend their results by providing a stronger approximation algorithm for this latter problem. 1
Approximating minimum cost connectivity problems
 58 in Approximation algorithms and Metaheuristics, Editor
, 2007
"... ..."
Approximation Algorithms for MinimumCost kVertex Connected Subgraphs
 In 34th Annual ACM Symposium on the Theory of Computing
, 2002
"... We present two new algorithms for the problem of nding a minimumcost kvertex connected spanning subgraph. The rst algorithm works on undirected graphs with at least 6k vertices and achieves an approximation of 6 times the kth harmonic number (which is O(log k)), The second algorithm works o ..."
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Cited by 69 (2 self)
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We present two new algorithms for the problem of nding a minimumcost kvertex connected spanning subgraph. The rst algorithm works on undirected graphs with at least 6k vertices and achieves an approximation of 6 times the kth harmonic number (which is O(log k)), The second algorithm works on any graph (directed or undirected) and gives an O( n=)approximation algorithm for any > 0 and k (1 )n. These algorithms improve on the previous best approximation factor (more than k=2). The latter algorithm also extends to other problems in network design with vertex connectivity requirements. Our main tools are setpair relaxations, a theorem of Mader's (in the undirected case) and iterative rounding (general case).
An Improved LPbased Approximation for Steiner Tree
, 2009
"... The Steiner tree problem is one of the most fundamentalhard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum weight tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from to the current best���[Robin ..."
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Cited by 65 (7 self)
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The Steiner tree problem is one of the most fundamentalhard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum weight tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from to the current best���[Robins,ZelikovskySIDMA’05]. All these algorithms are purely combinatorial. A longstanding open problem is whether there is an LPrelaxation for Steiner tree with integrality gap smaller than [Vazirani,RajagopalanSODA’99]. In this paper we improve the approximation factor for Steiner tree, developing an LPbased approximation a� algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directedcomponent cut relaxation for the�restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components together
Approximating Minimum Bounded Degree Spanning Trees to within One of
 In Proceedings of the 33rd International Colloquium on Automata, Languages and Programming.
, 2006
"... In the Minimum Bounded Degree Spanning Tree problem, we are given an undirected graph G = (V, E) with a degree upper bound B v on each vertex v ∈ V , and the task is to find a spanning tree of minimum cost that satisfies all the degree bounds. Let OPT be the cost of an optimal solution to this prob ..."
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Cited by 63 (7 self)
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In the Minimum Bounded Degree Spanning Tree problem, we are given an undirected graph G = (V, E) with a degree upper bound B v on each vertex v ∈ V , and the task is to find a spanning tree of minimum cost that satisfies all the degree bounds. Let OPT be the cost of an optimal solution to this problem. In this article we present a polynomialtime algorithm which returns a spanning tree T of cost at most OPT
Survivable network design with degree or order constraints
 SIAM J. ON COMPUTING
, 2009
"... We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requir ..."
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Cited by 61 (7 self)
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We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds Bv on the vertices. This includes the wellstudied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a (2, 2Bv +3)approximation algorithm for the edgeconnectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex v is at most 2Bv + 3. This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly kEdgeConnected Subgraph problem. In contrast, we show that the vertexconnectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic
Approximation Algorithms for NonUniform BuyatBulk Network Design
, 2006
"... We consider approximation algorithms for nonuniform buyatbulk network design problems. The first nontrivial approximation algorithm for this problem is due to Charikar and Karagiozova (STOC' 05); for an instance on h pairs their algorithm has an approximation guarantee of exp( O(plog h log ..."
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Cited by 55 (12 self)
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We consider approximation algorithms for nonuniform buyatbulk network design problems. The first nontrivial approximation algorithm for this problem is due to Charikar and Karagiozova (STOC' 05); for an instance on h pairs their algorithm has an approximation guarantee of exp( O(plog h log log h)) for the uniformdemand case, and log D * exp(O(plog h log log h)) for the general demand case, where D is the total demand. We improve upon this result, by presenting the first polylogarithmic approximation for this problem. The ratio we obtain is O(log3 h * min{log D, fl(h2)}) where h is the number of pairs and fl(n) is the worst case distortion in embedding the metric induced by a n vertex graph into a distribution over its spanning trees. Using the best known upper bound on fl(n) we obtain an O(min{log3 h*log D, log5 h log log h})ratio approximation. We also give polylogarithmic approximations for some variants of the singesource problem that we need for the multicommodity problem.