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24
Potential Function Methods for Approximately Solving Linear Programming Problems: Theory and Practice
, 2001
"... After several decades of sustained research and testing, linear programming has evolved into a remarkably reliable, accurate and useful tool for handling industrial optimization problems. Yet, large problems arising from several concrete applications routinely defeat the very best linear programming ..."
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Cited by 69 (4 self)
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After several decades of sustained research and testing, linear programming has evolved into a remarkably reliable, accurate and useful tool for handling industrial optimization problems. Yet, large problems arising from several concrete applications routinely defeat the very best linear programming codes, running on the fastest computing hardware. Moreover, this is a trend that may well continue and intensify, as problem sizes escalate and the need for fast algorithms becomes more stringent. Traditionally, the focus in optimization algorithms, and in particular, in algorithms for linear programming, has been to solve problems "to optimality." In concrete implementations, this has always meant the solution ofproblems to some finite accuracy (for example, eight digits). An alternative approach would be to explicitly, and rigorously, trade o# accuracy for speed. One motivating factor is that in many practical applications, quickly obtaining a partially accurate solution is much preferable to obtaining a very accurate solution very slowly. A secondary (and independent) consideration is that the input data in many practical applications has limited accuracy to begin with. During the last ten years, a new body ofresearch has emerged, which seeks to develop provably good approximation algorithms for classes of linear programming problems. This work both has roots in fundamental areas of mathematical programming and is also framed in the context ofthe modern theory ofalgorithms. The result ofthis work has been a family ofalgorithms with solid theoretical foundations and with growing experimental success. In this manuscript we will study these algorithms, starting with some ofthe very earliest examples, and through the latest theoretical and computational developments.
Capacitated Network Design  Polyhedral Structure and Computation
 INFORMS JOURNAL ON COMPUTING
, 1994
"... We study a version of the capacity expansion problem (CEP) that arises in telecommunication network design. Given a capacitated network and a traffic demand matrix, the objective in the CEP is to add capacity to the edges, in batches of various modularities, and route traffic, so that the overall co ..."
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Cited by 59 (8 self)
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We study a version of the capacity expansion problem (CEP) that arises in telecommunication network design. Given a capacitated network and a traffic demand matrix, the objective in the CEP is to add capacity to the edges, in batches of various modularities, and route traffic, so that the overall cost is minimized. We study the polyhedral structure of a mixedinteger formulation of the problem and develop a cuttingplane algorithm using facet defining inequalities. The algorithm produces an extended formulation providing both a very good lower bound and a starting point for branch and bound. The overall algorithm appears effective when applied to problem instances using reallife data.
Minimum cost capacity installation for multicommodity network flows
 MATHEMATICAL PROGRAMMING
, 1998
"... Consider a directed graph G = (V; A), and a set of traffic demands to be shipped between pairs of nodes in V. Capacity has to be installed on the edges of this graph (in integer multiples of a base unit) so that traffic can be routed. In this paper we consider the problem of minimum cost installatio ..."
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Cited by 49 (12 self)
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Consider a directed graph G = (V; A), and a set of traffic demands to be shipped between pairs of nodes in V. Capacity has to be installed on the edges of this graph (in integer multiples of a base unit) so that traffic can be routed. In this paper we consider the problem of minimum cost installation of capacity on the arcs to ensure that the required demands can be shipped simultaneously between node pairs. We study two different approaches for solving problems of this type. The first one is based on the idea of metric inequalities (see Onaga and Kakusho[1971]), and uses a formulation with only jAj variables. The second uses an aggregated multicommodity flow formulation and has jV j \Delta jAj variables. We first describe two classes of strong valid inequalities and use them to obtain a complete polyhedral description of the associated polyhedron for the complete graph on 3 nodes. Next we explain our solution methods for both of the approaches in detail and present computational results. Our computational experience shows that the two formulations are comparable and yield effective algorithms for solving reallife problems.
A BranchandCut Algorithm for Capacitated Network Design Problems
 MATHEMATICAL PROGRAMMING
, 1998
"... We present a branchandcut algorithm to solve capacitated network design problems. Given a capacitated network and pointtopoint traffic demands, the objective is to install more capacity on the edges of the network and route traffic simultaneously, so that the overall cost is minimized. We study ..."
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Cited by 32 (2 self)
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We present a branchandcut algorithm to solve capacitated network design problems. Given a capacitated network and pointtopoint traffic demands, the objective is to install more capacity on the edges of the network and route traffic simultaneously, so that the overall cost is minimized. We study a mixedinteger programming formulation of the problem and identify some new facet defining inequalities. These inequalities, together with other known combinatorial and mixedinteger rounding inequalities, are used as cutting planes. To choose the branching variable, we use a new rule called "knapsack branching". We also report on our computational experience using reallife data.
Strong Inequalities for Capacitated Survivable Network Design Problems
 MATHEMATICAL PROGRAMMING
, 1999
"... We present several classes of facetdefining inequalities to strengthen polyhedra arising as subsystems of network design problems with survivability constraints. These problems typically involve assigning capacities to a network with multicommodity demands, such that after a vertex or edgedeletio ..."
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Cited by 31 (5 self)
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We present several classes of facetdefining inequalities to strengthen polyhedra arising as subsystems of network design problems with survivability constraints. These problems typically involve assigning capacities to a network with multicommodity demands, such that after a vertex or edgedeletion at least some prescribed fraction of each demand can be routed.
Costefficient network synthesis from leased lines
, 1997
"... Given a communication demand between each pair of nodes of a network we consider the problem of deciding what capacity to install on each edge of the network in order to minimize the building cost of the network and to satisfy the demand between each pair of nodes. The feasible capacities that can b ..."
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Cited by 16 (2 self)
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Given a communication demand between each pair of nodes of a network we consider the problem of deciding what capacity to install on each edge of the network in order to minimize the building cost of the network and to satisfy the demand between each pair of nodes. The feasible capacities that can be leased from a network provider are of a particular kind in our case. There are a few socalled basic capacities having the property that every basic capacity is an integral multiple of every smaller basic capacity. An edge can be equipped with a capacity only if it is an integer combination of the basic capacities. We treat, in addition, several restrictions on the routings of the demands (length restriction, diversification) and failures of single nodes or single edges. We formulate the problem as a mixed integer linear programming problem and develop a cutting plane algorithm as well as several heuristics to solve it. We report on computational results for real world data.
Design of Survivable Networks: A survey
 In Networks
, 2005
"... For the past few decades, combinatorial optimization techniques have been shown to be powerful tools for formulating and solving optimization problems arising from practical situations. In particular, many network design problems have been formulated as combinatorial optimization problems. With the ..."
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Cited by 14 (0 self)
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For the past few decades, combinatorial optimization techniques have been shown to be powerful tools for formulating and solving optimization problems arising from practical situations. In particular, many network design problems have been formulated as combinatorial optimization problems. With the advances of optical technologies and the explosive growth of the Internet, telecommunication networks have seen an important evolution and therefore, designing survivable networks has become a major objective for telecommunication operators. Over the past years, a big amount of research has then been done for devising efficient methods for survivable network models, and particularly cutting plane based algorithms. In this paper, we attempt to survey some of these models and the optimization methods used for solving them.
Capacity and survivability models for telecommunication networks
 in Proceedings of EURO/INFORMS Meeting
, 1997
"... Designing lowcost networks that survive certain failure situations is one of the prime tasks in the telecommunication industry. In this paper we survey the development of models for network survivability used in practice in the last ten years. We show how algorithms integrating polyhedral combinato ..."
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Cited by 13 (1 self)
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Designing lowcost networks that survive certain failure situations is one of the prime tasks in the telecommunication industry. In this paper we survey the development of models for network survivability used in practice in the last ten years. We show how algorithms integrating polyhedral combinatorics, linear programming, and various heuristic ideas can help solve realworld network dimensioning instances to optimality or within reasonable quality guarantees in acceptable running times. The most general problem type we address is the following. Let a communication demand between each pair of nodes of a telecommunication network be given. We consider the problem of choosing, among a discrete set of possible capacities, which capacity to install on each of the possible edges of the network in order to (i) satisfy all demands, (ii) minimize the building cost of the network. In addition to determining the network topology and the edge capacities we have to provide, for each demand, a routing such that (iii) no path can carry more than a given percentage of the demand, (iv) no path in the routing exceeds a given length. We also have to make sure that (v) for every single node or edge failure, a certain percentage of the demand is reroutable. Moreover, for all failure situations feasible routings must be computed. The model described above has been developed in cooperation with a German mobile phone provider. We present a mixedinteger programming formulation of this model and computational results with data from practice.
Experiments With a Network Design Algorithm Using EpsilonApproximate Linear Programs
, 1998
"... We describe an upperbound algorithm for multicommodity network design problems that relies on new results for approximately solving certain linear programs, and on the greedy heuristic for setcovering problems. 1 Introduction. Network design problems are mixedinteger programs that have the fo ..."
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Cited by 8 (3 self)
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We describe an upperbound algorithm for multicommodity network design problems that relies on new results for approximately solving certain linear programs, and on the greedy heuristic for setcovering problems. 1 Introduction. Network design problems are mixedinteger programs that have the following broad structure. Given a graph, and a set of "demands"  positive amounts to be routed between pairs of vertices  capacity must be added to the edges and/or vertices of the graph, in discrete amounts, and at minimum cost, so that a feasible routing is possible. Problem of this form are increasingly important in telecommunications applications, because of the great expense inherent in maintaining and upgrading metropolitan networks. A wide variety of special cases have been studied. For example, one may be constrained to using a fixed family of paths to carry out the routing, or to using a single path for each demand, or to using integral flows. The precise manner in which capacit...