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THE PRIMALDUAL METHOD FOR APPROXIMATION ALGORITHMS AND ITS APPLICATION TO NETWORK DESIGN PROBLEMS
"... The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent researc ..."
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Cited by 123 (7 self)
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The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent research applying the primaldual method to problems in network design.
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 80 (12 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 62 (1 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).
Strengthening Integrality Gaps for Capacitated Network Design and Covering Problems
"... A capacitated covering IP is an integer program of the form min{cxUx ≥ d, 0 ≤ x ≤ b, x ∈ Z +}, where all entries of c, U, and d are nonnegative. Given such a formulation, the ratio between the optimal integer solution and the optimal solution to the linear program relaxation can be as bad as d∞ ..."
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Cited by 61 (1 self)
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A capacitated covering IP is an integer program of the form min{cxUx ≥ d, 0 ≤ x ≤ b, x ∈ Z +}, where all entries of c, U, and d are nonnegative. Given such a formulation, the ratio between the optimal integer solution and the optimal solution to the linear program relaxation can be as bad as d∞, even when U consists of a single row. We show that by adding additional inequalities, this ratio can be improved significantly. In the general case, we show that the improved ratio is bounded by the maximum number of nonzero coefficients in a row of U, and provide a polynomialtime approximation algorithm to achieve this bound. This improves the previous best approximation algorithm which guaranteed a solution within the maximum row sum times optimum. We also show that for particular instances of capacitated covering problems, including the minimum knapsack problem and the capacitated network design problem, these additional inequalities yield even stronger improvements in the IP/LP ratio. For the minimum knapsack, we show that this improved ratio is at most 2. This is the first nontrivial IP/LP ratio for this basic problem. Capacitated network design generalizes the classical network design problem by introducing capacities on the edges, whereas previous work only considers the case when all capacities equal 1. For capacitated network design problems, we show that this improved ratio depends on a parameter of the graph, and we also provide polynomialtime approximation algorithms to match this bound. This improves on the best previous mapproximation, where m is the number of edges in the graph. We also discuss improvements for some other special capacitated covering problems, including the fixed charge network flow problem. Finally, for the capacitated network design problem, we give some stronger results and algorithms for series parallel graphs and strengthen these further for outerplanar graphs. Most of our approximation algorithms rely on solving a single LP. When the original LP (before adding our strengthening inequalities) has a polynomial number of constraints, we describe a combinatorial FPTAS for the LP with our (exponentiallymany) inequalities added. Our contribution here is to describe an appropriate
Approximation algorithms for nonuniform buyatbulk network design problems
 Proc. of IEEE FOCS
"... Abstract. Buyatbulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given multicommodity flow demand between node pairs. We present approximation algorith ..."
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Cited by 58 (15 self)
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Abstract. Buyatbulk network design problems arise in settings where the costs for purchasing or installing equipment exhibit economies of scale. The objective is to build a network of cheapest cost to support a given multicommodity flow demand between node pairs. We present approximation algorithms for buyatbulk network design problems with costs on both edges and nodes of an undirected graph. Our main result is the first polylogarithmic approximation ratio for the nonuniform problem that allows different cost functions on each edge and node; the ratio we achieve is O(log4 h) where h is the number of demand pairs. In addition we present an O(log h) approximation for the single sink problem. Polylogarithmic ratios for some related problems are also obtained. Our algorithm for the multicommodity problem is obtained via a reduction to the single source problem using the notion of junction trees. We believe that this presents a simple yet useful general technique for network design problems. Key words. Nonuniform buyatbulk, network design, approximation algorithm, concave cost, network flow, economies of scale AMS subject classifications. 68Q25, 68W25, 90C27, 90C59 1. Introduction. Network
Survivable network design with degree or order constraints
 SIAM J. on Computing
, 2009
"... Abstract. 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 connectivit ..."
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Cited by 55 (8 self)
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Abstract. 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
Hardness of Approximation for VertexConnectivity NetworkDesign Problems
, 2002
"... In the survivable network design problem (SNDP), the goal is to find a minimumcost spanning subgraph satisfying certain connectivity requirements. We study the vertexconnectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertexdisjoint paths con ..."
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Cited by 44 (5 self)
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In the survivable network design problem (SNDP), the goal is to find a minimumcost spanning subgraph satisfying certain connectivity requirements. We study the vertexconnectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertexdisjoint paths connecting them.
An Iterative Rounding 2Approximation Algorithm for the Element Connectivity Problem
 In 42nd Annual IEEE Symposium on Foundations of Computer Science
, 2001
"... In the edge connected version of this problem (ECSNDP), these paths must be edgedisjoint. In the vertex connected version of the problem (VCSNDP), the paths must be vertex disjoint. Jain et al. [12] propose a version of the problem intermediate in difficulty to these two, called the element conne ..."
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Cited by 32 (3 self)
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In the edge connected version of this problem (ECSNDP), these paths must be edgedisjoint. In the vertex connected version of the problem (VCSNDP), the paths must be vertex disjoint. Jain et al. [12] propose a version of the problem intermediate in difficulty to these two, called the element connectivity problem (ELCSNDP, or ELC). In this problem, the set of vertices is partitioned into terminals and nonterminals. The edges and nonterminals of the graph are called elements. The values are only specified for pairs of terminals must be element disjoint. Thus if are still connected by a path in the network. These variants of SNDP are all known to be NPhard. The best known approximation algorithm for the ECSNDP has performance guarantee of 2 (due to Jain [11]), and iteratively rounds solutions to a linear programming relaxation of the problem. ELC has a primaldual  approximation algorithm, where (Jain et al. [12]). VCSNDP is not known to have a nontrivial approximation algorithm; however, recently Fleischer [7] has shown how to extend the technique of Jain [11] to give a 2approximation algorithm in the case that ! . She also shows that the same techniques will not work for VCSNDP for more general values of . In this paper we show that these techniques can be extended to a 2approximation algorithm for ELC. This gives the first constant approximation algorithm for a general survivable network design problem which allows node failures.
Iterative Rounding 2Approximation Algorithms for MinimumCost Vertex Connectivity Problems
 J. Comput. Syst. Sci
, 2002
"... The survivable network design problem (SNDP) is the following problem: given an undirected graph and values r ij for each pair of vertices i and j, find a minimumcost subgraph such that there are r ij disjoint paths between vertices i and j. In the edge connected version of this problem (ECSNDP) ..."
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Cited by 31 (0 self)
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The survivable network design problem (SNDP) is the following problem: given an undirected graph and values r ij for each pair of vertices i and j, find a minimumcost subgraph such that there are r ij disjoint paths between vertices i and j. In the edge connected version of this problem (ECSNDP) , these paths must be edgedisjoint. In the vertex connected version of the problem (VCSNDP), the paths must be vertex disjoint. The element connectivity problem (ELCSNDP, or ELC) is a problem of intermediate difficulty.