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A general approximation technique for constrained forest problems
 in Proceedings of the 3rd Annual ACMSIAM Symposium on Discrete Algorithms
, 1992
"... Abstract. We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles, or paths satisfying certain requirements. In particular, many basic combinatorial optimizatio ..."
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Cited by 355 (21 self)
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Abstract. We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles, or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimumcost spanning tree, minimumweight perfect matching, traveling salesman, and Steiner tree problems. Our technique produces approximation algorithms that run in O(n log n) time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2approximation algorithm for the minimumweight perfect matching problem under the triangle inequality. Our running time of O(n log n) time compares favorably with the best strongly polynomial exact algorithms running in O(n 3) time for dense graphs. A similar result is obtained for the 2matching problem and its variants. We also derive the first approximation algorithms for many NPcomplete problems, including the nonfixed pointtopoint connection problem, the exact path partitioning problem, and complex locationdesign problems. Moreover, for the prizecollecting traveling salesman or Steiner tree problems, we obtain 2approximation algorithms, therefore improving the previously bestknown performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413420].
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
Improved Approximation Algorithms for Network Design Problems
, 1994
"... We consider a class of network design problems in which one needs to find a minimumcost network satisfying certain connectivity requirements. For example, in the survivable network design problem, the requirements specify that there should be at least r(v; w) edgedisjoint paths between each pai ..."
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Cited by 80 (11 self)
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We consider a class of network design problems in which one needs to find a minimumcost network satisfying certain connectivity requirements. For example, in the survivable network design problem, the requirements specify that there should be at least r(v; w) edgedisjoint paths between each pair of vertices v and w. We present an approximation algorithm with a performance guarantee of 2H(fmax ) = 2(1 + 2 + 3 + \Delta \Delta \Delta + fmax ) where fmax is the maximum requirement. This improves upon the best previously known performance guarantee of 2fmax . We also show
An Efficient Approximation Algorithm for the Survivable Network Design Problem
 IN PROCEEDINGS OF THE THIRD MPS CONFERENCE ON INTEGER PROGRAMMING AND COMBINATORIAL OPTIMIZATION
, 1993
"... The survivable network design problem is to construct a minimumcost subgraph satisfying certain given edgeconnectivity requirements. The first polynomialtime approximation algorithm was given by Williamson et al. [20]. This paper gives an improved version that is more efficient. Consider a graph ..."
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Cited by 50 (7 self)
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The survivable network design problem is to construct a minimumcost subgraph satisfying certain given edgeconnectivity requirements. The first polynomialtime approximation algorithm was given by Williamson et al. [20]. This paper gives an improved version that is more efficient. Consider a graph
A Fast Approximation Scheme for Fractional Covering Problems with Box Constraints
, 2004
"... We present the first combinatorial approximation scheme that yields a pure approximation guarantee for linear programs that are either covering problems with upper bounds on variables, or their duals. Existing approximation schemes for mixed covering and packing problems do not simultaneously satis ..."
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Cited by 18 (2 self)
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We present the first combinatorial approximation scheme that yields a pure approximation guarantee for linear programs that are either covering problems with upper bounds on variables, or their duals. Existing approximation schemes for mixed covering and packing problems do not simultaneously satisfy packing and covering constraints exactly. We present the first combinatorial approximation scheme that returns solutions that simultaneously satisfy general positive covering constraints and upper bounds on variable values. For input parameter ffl? 0, the returned solution has positive linear objective function value at most 1 + ffl times the optimal value. The general algorithm requires O(ffl2m log(cTu)) iterations, where c is the objective cost vector, u is the vector of upper bound values, and m is the number of variables. Each iteration uses an oracle that finds an (approximately) most violated constraint. A natural set of problems that our work addresses are linear programs for various network design problems: generalized Steiner network, vertex connectivity, directed connectivity, capacitated network design, group Steiner forest. The integer versions of these problems are all NPhard. For each of them, there is an approximation algorithm that rounds the solution to the corresponding linear program relaxation. If the LP solution is not feasible, then the corresponding integer solution will also not be feasible. Solving the linear program is often the computational bottleneck in these problems, and thus a fast approximation scheme for the LP relaxation means faster approximation algorithms. For these applications, we introduce a new modification of the pushrelabel maximum flow algorithm that allows us to perform each iteration in amortized O(jEj+jV j log jV j) time, instead of one maximum flow per iteration that is implied by the straight forward adaptation of our general algorithm. In conjunction with an observation that reduces the number of iterations to jEj log jV j for f0; 1g constraint matrices, the modification allows us to obtain an algorithm that is faster than existing exact or approximate algorithms by a factor of at least O(jEj) and by a factor of O(jEj log jV j) if the number of demand pairs is \Omega (jV j).
From Valid Inequalities to Heuristics: A Unified View of Primaldual Approximation Algorithms in CoveringProblems
 Operations Research
, 1998
"... In recent years approximation algorithms based on primaldual methods have been successfully applied to a broad class of discrete optimization problems. In this paper, we propose a generic primaldual framework to design and analyze approximation algorithms for integer programming problems of the co ..."
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Cited by 12 (0 self)
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In recent years approximation algorithms based on primaldual methods have been successfully applied to a broad class of discrete optimization problems. In this paper, we propose a generic primaldual framework to design and analyze approximation algorithms for integer programming problems of the covering type that uses valid inequalities in its design. The worstcase bound of the proposed algorithm is related to a fundamental relationship (called strength) between the set of valid inequalities and the set of minimal solutions to the covering problems. In this way, we can construct an approximation algorithm simply by constructing the required valid inequalities. We apply the proposed algorithm to several problems, such as covering problems related to totally balanced matrices, cyclic scheduling, vertex cover, general set covering, intersections of polymatroids, and several network design problems attaining (in most cases) the best worstcase bound known in the literature. In the last 20 years, two approaches to discrete optimization problems have emerged: polyhedral combinatorics and approximation algorithms. Under the first approach, researchers formulate problems as integer programs and solve their linear programming relaxations. By adding strong valid inequalities (preferably facets of the convex hull of solutions) to enhance the formulations, researchers
Approximation Algorithms for MinMax Tree Partition
, 1997
"... We consider the problem of partitioning the node set of a graph into p equal sized subsets. The objective is to minimize the maximum length, over these subsets, of a minimum spanning tree. We show that no polynomial algorithm with bounded Ž 2 error ratio can be given for the problem unless P � NP. W ..."
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Cited by 5 (1 self)
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We consider the problem of partitioning the node set of a graph into p equal sized subsets. The objective is to minimize the maximum length, over these subsets, of a minimum spanning tree. We show that no polynomial algorithm with bounded Ž 2 error ratio can be given for the problem unless P � NP. We present an On. time algorithm for the problem, where n is the number of nodes in the graph. Assuming that the edge lengths satisfy the triangle inequality, its error ratio is at most 2 p � 1. We also present an improved algorithm that obtains as an input a positive Ž Ž p�x. p 2 integer x. It runs in O 2 n. time, and its error ratio is at most Ž2�x� Ž x�p�1.. p.
Steiner Trees and Beyond: Approximation Algorithms for Network Design
, 1993
"... We present approximation algorithms for several NPhard optimization problems arising in network design. Almost all of our algorithms run in polynomial time and output solutions with a worstcase performance guarantee on the quality of the output solution. A typical problem that we consider can be s ..."
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Cited by 3 (1 self)
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We present approximation algorithms for several NPhard optimization problems arising in network design. Almost all of our algorithms run in polynomial time and output solutions with a worstcase performance guarantee on the quality of the output solution. A typical problem that we consider can be stated as follows: given an undirected graph and certain connectivity requirements, find a subgraph that satisfies these requirements and has minimum cost. In this thesis, we address many different connectivity requirements such as spanning trees, Steiner trees, generalized Steiner forests, and twoconnected networks. The cost criteria that we consider range from the total cost of the edges in the network, the total cost of the nodes in the network, the maximum degree of any node in the network, the maximum cost of any edge in the network to some combination of these. We also address the maximumleaf spanning tree problem and provide the first approximation algorithms for this problem. In t...
Connectivity Augmentation
, 1994
"... The problem of connectivity augmentation consists of finding a minimum cost of new edges to be added to a given graph so as to satisfy some prescribed connectivity requirements. This paper surveys cases when polynomial time algorithms and/or good characterizations are available for the minimum. 1. ..."
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The problem of connectivity augmentation consists of finding a minimum cost of new edges to be added to a given graph so as to satisfy some prescribed connectivity requirements. This paper surveys cases when polynomial time algorithms and/or good characterizations are available for the minimum. 1. INTRODUCTION In network design it is a fundamental problem to construct graphs or subgraphs of a graph of minimum cost satisfying certain connectivity specifications. Shortest paths between two specified nodes, or minimum cost spanning trees may be viewed as (wellknown) special cases of this problem. Very often a starting graph is already available and the goal is to augment the graph. For example, at least how many new edges must be added to a digraph to make it strongly connected? Having such a broad class of problems (already a special case, the wellknown Steinertree problem, has a vast literature), it is of no surprise that a large number of connectivity augmentation problems is NP...