D. Bienstock, S. Chopra, O. Gunluk, C. Tsai, Minimum cost capacity installation for multicommodity network flows, Math. Programming B 81 2 (1998) 177--199.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....links cost a few hundred dollars. As a result, the existing research in network design proved less applicable than we had hoped. The SAN fabric design problem generalizes and extends several NP hard problems in the network design literature, including the multicommodity network design problem [8,3,2,7,9,5,1], which involves choosing a minimum cost set of capacitated, fixed cost links to connect a known set of nodes in order to satisfy multicommodity flow requirements. This problem is known to be NP hard even in the single commodity case, and is notoriously difficult to solve in practice. Unlike in ....

D. Bienstock, S. Chopra and O. Gunluk, Minimum cost capacity installation for multicommodity network flows, Mathematical Programming 81 (1998), no. 2-1, 177-199

....for these cut inequalities is the max cut problem, which is NP hard, but for the problem instances reported the separation problem can be solved relatively easily. The closely related minimum cost capacity installation problem for multi commodity flows has been studied in Bienstock et al. BCGT98] where capacities can be selected in multiples of one level. A multicommodity flow formulation and a capacity formulation is suggested. Several valid facet defining inequalities are given for the different formulations, and a branch and cut algorithm is suggested. An integer multicommodity flow ....

D. Bienstock, S. Chopra, O. GSnlSk, and C.-Y. Tsai. Minimum cost capacity installation for multicommodity network flows. Math. Program., 81(2), 177-200, 1998.

....[23] for a survey of work on the Steiner tree problem. The nonbifurcated network loading problem is NP hard even when all commodities share a single source [21] If we relax the constraint that flows cannot be split, the SAN design problem generalizes the multicommodity network design problem [20, 8, 7, 19, 22, 10, 5]. This problem is known to be NP hard even in the single commodity case [15] Like the nonbifurcated network loading problem, it involves choosing a set of capacitated, fixed cost links to connect a set of nodes to satisfy multicommodity flow requirements. Any number of links between a pair of ....

D. Bienstock, S. Chopra, and O. Gunluk, Minimum cost capacity installation for multicommodity network flows, Mathematical Programming 81 (1998), no. 2--1, 177--199.

....in the general systems cited above significantly improve performance on some instances. In contrast there has been considerable work on multi commodity problems arising from telecommunications networks. Among others single arc sets [70] and MIR inequalities [26] have been used, and both heuristics [20, 21], total enumeration [20] and max cut [16] have been used to generate good cut sets. See also [6, 38, 39, 57] Lot Sizing, Facility Location and other Structured MIPs A variety of multi item and multi level lot sizing problems have been solved using the cutting planes described above, see [32, 92, ....

D. Bienstock, S. Chopra, O. Gunluk, and C-Y. Tsai, Minimum cost capacity installation for multicommodity network flows, Mathematical Programming 81, 177 -- 199 (1998).

....problems with real life data for instances of up to 16 nodes to optimality. Their model includes flow costs, and the capacities can be chosen as combinations of two basic technologies (OC3 and OC12 facilities) In another study with one basic technology Bienstock, Chopra, Gunluk, and Tsai (see [4]) solve New York area problems with up to 15 nodes and Norwegian Problems with up to 27 nodes (supplied by M. Stoer) almost to optimality. Magnanti, Mirchandani, and Vachani [12] investigate the same problem without flow costs and solve randomly generated instances with up to 15 nodes with ....

D. Bienstock, S. Chopra, O. Gunluck, and C.-Y. Tsai. Minimum cost capacity installation for multicommodity network flows. Mathematical Programming (Special Issue on Computational Integer Programming), (to appear), 1995.

....survivability requirements. IBM Corporation, Philadelphia, USA, alevras us.ibm.com y Konrad Zuse Zentrum fur Informationstechnik, Berlin, Germany, groetschel zib.de z Konrad Zuse Zentrum fur Informationstechnik, Berlin, Germany, wessaely zib.de 1 This problems has many versions; see, e.g. [1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 18] to mention a few relevant references. Of course, the link capacities must be chosen in a such way that all demands can be satisfied. An important design aspect is the protection of the network against component failures and the handling of such situations. Our partner E Plus considers three ....

D. Bienstock, S. Chopra, O. Gunluck, and C.-Y. Tsai. Minimum cost capacity installation for multicommodity network flows. Mathematical Programming (Special Issue on Computational Integer Programming), (to appear), 1995.

....the first type. Pochet and Wolsey (1995) study the polyhedron of a single arc network design problem with an arbitrary number of facilities with divisible integer capacities. Chopra et al. 1998) give new inequalities and extended formulations for single commodity problems with two facility types. Bienstock et al. 1998) consider a multicommodity single facility network design problem. 2 All of the prior studies on the problem assume that facility capacities are integer multiples of some basic capacity unit. Even though such an assumption is valid for some applications, this is not always the case for ....

Bienstock, D., Chopra, S., Gunluk, O., and Tsai, C.-Y. (1998). Minimum cost capacity installation for multicommodity networks. Mathematical Programming, 81:177--199.

....in the general systems cited above significantly improve performance on some instances. In contrast there has been considerable work on multi commodity problems arising from telecommunications networks. Among others single arc sets [70] and MIR inequalities [26] have been used, and both heuristics [20, 21], total enumeration [20] and max cut [16] have been used to generate good cut sets. See also [6, 38, 39, 57] Lot Sizing, Facility Location and other Structured MIPs A variety of multi item and multi level lot sizing problems have been solved using the cutting planes described above, see [32, 92, ....

D. Bienstock, S. Chopra, O. Gunluk, and C-Y. Tsai, Minimum cost capacity installation for multicommodity network flows, Mathematical Programming 81, 177 -- 199 (1998).

....unidirectional and bidirectional capacity usage, i.e. if an edge contains a unit of capacity this unit can either be used in one or in both directions of the edge. In most studies the unidirectional (or undirected) case is studied. Exceptions are Bienstock and G unl uk [6] and Bienstock et al. [5], who study the bidirectional case. In the sequel we consider both possibilities. Moreover, we show that the models of the corresponding NLPs have many common aspects. With respect to network lay out we do not specialize us. For instance, we do not take reliability requirements into account. For ....

D. Bienstock, S. Chopra, O. Gunluk, and C.-Y. Tsai. Minimum cost capacity installation for multicommodity network ows. Mathematical Programming, 81:177-199, July 1998. Working paper.

....gap between heuristic solutions and Lagrangian lower bounds. Due to this, even small instances of the problem cannot be solved to anywhere near optimality by state of the art computational techniques. More recent study of our network design problem with multiple sinks is undertaken in [BG 96, BCGT 98, Bar 96] These papers develop cutting plane methods by exploiting classes of valid inequalities for an appropriate formulation that only considers one or two cable types. Mansour and Peleg [MP 94] have results on a variant of our problem. In their model, there are multiple sinks and multiple ....

D. Bienstock, S. Chopra, O. Gunluk and C-Y. Tsai, "Minimum cost capacity installation for multicommodity network flows,", Mathematical Programming, 81, 177-199, 1998.

....paths between any pair of nodes never exceed two) All these survivable problems are uncapacitated, except the model studied by Stoer and Dahl [61] which also uses a cut based approach. More recently, researchers have studied cut based models for more general capacitated network design problems [13, 16, 57]. Network design problems on trees are also amenable to so called packing based formulations [54] All these modeling approaches can handle additional requirements that are often encountered in practice. We give a brief list of some of these requirements, with pointers to the literature indicating ....

Bienstock D., Chopra S., Gunluk O. and Tsai C.-Y. (1995), Minimum-Cost Capacity Installation for Multicommodity Network Flows, working paper, Department of Industrial Engineering and Operations Research, Columbia University.

....resilience, capacity reservation. 1 Introduction 1.1 Overview A commonly encountered network design problem is that of reserving capacities in a network so as to support some given traffic matrix. Algorithms for this problem have been developed by a number of groups, see for example [5] 8] [7], 20] 21] 22] These are primarily based on polyhedral methods and will often have the ability to produce optimal solutions together with certificates. Perhaps surprisingly, many network planning tools for this problem seem to be based on successively solving a shortest path problem for each ....

D. Bienstock, S. Chopra, O. Gunlik, C. Tsai, Minimum cost capacity installation for multicommodity network flows, Math. Programming B 81 2 (1998) 149--176.

....gap between heuristic solutions and Lagrangian lower bounds. Due to this, even small instances of the problem cannot be solved to anywhere near optimality by state of the art computational techniques. More recent study of our network design problem with multiple sinks is undertaken in [BG 95, BCGT 95] These papers develop cutting plane methods by exploiting classes of valid inequalities for an appropriate formulation that only considers one or two cable types. Mansour and Peleg [MP 94] have results on a variant of our problem. In their model, there are multiple sinks and multiple sources, ....

D. Bienstock, S. Chopra, O. Gunluk and C-Y. Tsai, "Minimum cost capacity installation for multicommodity network flows,", manuscript, July 1995.

....in the resulting network so that the total capacity installation plus routing cost is minimized. Capacity can be installed in discrete multiples of various modularities. Polyhedral structure of this problem has been studied in [6] and closely related problems have been studied in [2] 3] [5], 9] 19] and [21] A variant of this problem where the initial capacities are assumed to be zero is also known as the network loading problem. We note that CEP is strongly NP hard as it contains the fixed charge network design problem, and thus the Steiner tree problem as a special case. ....

....and P CEP have the same number of variables. As we aim to solve instances with dense traffic matrices, we use P CEP in the remainder of the paper. We also note that it is possible to project out the flow variables from P CEP to obtain a formulation in the space of the discrete variables only [5]. This formulation requires an exponential number of constraints and is discussed more thoroughly in Section 2. 1.2 Branch and Cut Branch and cut is a relatively recent but well accepted approach introduced by Padberg and Rinaldi ( 25] and [26] in the context of the traveling salesman problem. ....

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D. Bienstock, S. Chopra, O. Gunluk, and C-Y. Tsai, "Minimum Cost Capacity Installation for Multicommodity Network Flows" to appear, Math. Programming.

....Commodity Capacitated Problem We now consider the two commodity one facility capacitated network design problem (TCOF) in which capacity can be purchased in batches of C units on each arc (i; j) 2 A at cost w ij 0. Flow costs are assumed to be zero. Magnanti, Mirchandani and Vachani (1995) and Chopra et.al. 1995) have studied the two facility version of the problem, where capacity is available in batches of 1 or C units. Chopra, Gilboa and Sastry (1996) studied the single origin destination version of the one and two facility problem where they describe an exact algorithm and an extended formulation for ....

Chopra, S., D.Bienstock, O.Gunluck, C.Y.Tsai, "Minimum cost capacity installation for multi commodity network flows", Research Report, Northwestern University, January 1995.

....arbitrary, 10) has the same coefficients as the strengthened cut inequality 8, hence it defines a facet. Note that condition (i) is also necessary for (8) to define a facet. Remark. The strengthened cut inequalities may be generalized in the spirit of the flow cutset inequalities introduced in [3]. Let F be a subset of ffi G (W ) In the validity proof above add the inequalities x k (ffi G (W ) as before, but now add the knapsack inequalities only for e 2 F . The resulting flowcutset inequality is y(F ) X k2K 0 x k (ffi G (W ) n F ) Gamma X k2K 2 x k (F ) djK 0 ....

D. Bienstock, S. Chopra, O. Gunluk, C-Y. Tsai, Minimum cost capacity installation for multicommodity network flows, draft, Columbia University, New York, January 1995.

....formulation by the user, and led us to develop an algorithmic approach adapted for use with a general purpose solver. Many related network design problems have been studied recently in which the capacity installation options differ slightly. Barahona [2] Bienstock and Gunluk [3] Bienstock et al.[4] and Magnanti, Mirchandani and Vachani[11] among others treat problems in which demands can be split among several paths. A variety of inequalities and separation algorithms have been developed both for families of valid inequalities of a global nature such as cut and strengthened cut inequalities ....

....Mirchandani and Vachani[11] among others treat problems in which demands can be split among several paths. A variety of inequalities and separation algorithms have been developed both for families of valid inequalities of a global nature such as cut and strengthened cut inequalities [2] 3] [4], 10] 11] partition inequalities [3] 4] 11] and flow cut set inequalities [3] and local edge cuts such as the residual capacity inequalities [11] Problems involving sending flow on a single path have been treated by Balakrishnan, Magnanti and Wong [1] for tree graphs, Gavish and Altinkemer ....

[Article contains additional citation context not shown here]

D. Bienstock, S. Chopra, O. Gunluk, and C-Y. Tsai, Minimum Cost Capacity Installation for Multicommodity Network Flows, Math. Programming B 81, 177-199 (1998).

....such that no more than a given fraction of the given demand is routed through any intermediate node. A large amount of work has been done by Minoux and others on the related model with a continuous cost function, see [12] and its references) 7] 1] A recent related model is studied in [3]. This paper is organized as follows. In Section 1 the integer lin2 ear programming model for the MULTISUN problem is presented. In addition it is explained how one obtains stronger LP relaxations by adding certain classes of valid inequalities originating from knapsacklike substructure of the ....

D. Bienstock, S. Chopra, O. G#nl#k and C.Y. Tsai, Minimum cost capacity installation for multicommodity network AEows. Draft. Columbia University, New York, January 1995.

.... problem with a single continuous variable can be obtained as MIR inequalities (see Marchand and Wolsey [13] For many other linear mixed integer models, simple MIR inequalities have been derived to produce strong valid inequalities which have proven to be computationally very effective (see [6] [7], 14] 18] and [20] Our primary objective in this paper is to contribute to the development of (general) techniques that can be used to generate new classes of strong valid inequalities for mixedinteger programs. More specifically, we investigate how to obtain new classes of valid inequalities ....

D. Bienstock, S. Chopra, O. Gunluk, and C-Y. Tsai, "Minimum Cost Capacity Installation for Multicommodity Network Flows" to appear in Mathematical Programming.

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D. Bienstock, S. Chopra, O. Gunluk and C. Tsai (1996), Minimum Cost Capacity Installation for Multicommodity Network Flows Math. Programming 81 (1998), 177-199. 17

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D. Bienstock, S. Chopra, O. Gunluk and C. Tsai, Minimum Cost Capacity Installation for Multicommodity Network Flows, Math. Programming 81 (1998), 177-199.

.... bandwidth capacity on each of the edges such that each demand can be routed on a single path from source to sink. The capacity on each edge is installed in discrete amounts and the associated costs are nonlinear. The total edge capacity installation costs are to be minimized. See [1] 2] [3], and [6] for related network design models. Applications of this problem (or its variants) arise in the telecommunications industry for both service providers (i.e. long distance carriers) and their customers. Our motivation for this study comes from the customer side of the problem where the ....

....(11) can be strengthened. 5 Cut set inequalities. In this section, we describe how the well known cut set inequalities can be strengthened using the special structure of our problem. These inequalities have been successfully applied to several capacitated network design problems (see [1] 2] [3] and [7] We consider the integer model (P ) 15 For any S V , let (S) E denote the set of edges with one end in S and the other in V n S and let K(S) fq 2 Q : jfi q ; j q g Sj = 1g be the set of demands that have to be routed across this cut set in any feasible solution. It is easy ....

[Article contains additional citation context not shown here]

D. Bienstock, S. Chopra, O. Gunluk, and C-Y. Tsai, \Minimum Cost Capacity Installation for Multicommodity Network Flows" to appear in Math. Programming.

.... problem with a single continuous variable can be obtained as MIR inequalities (see Marchand and Wolsey [13] For many other linear mixed integer models, simple MIR inequalities have been derived to produce strong valid inequalities which have proven to be computationally very e#ective (see [6] [7], 14] 18] and [20] Our primary objective in this paper is to contribute to the development of (general) techniques that can be used to generate new classes of strong valid inequalities for mixed integer programs. More specifically, we investigate how to obtain new classes of valid ....

D. Bienstock, S. Chopra, O. Gunluk, and C-Y. Tsai, "Minimum Cost Capacity Installation for Multicommodity Network Flows" to appear in Mathematical Programming.

....here does incorporate these reliability constraints. 4. SOLUTION PROCEDURE AND HEURISTICS The mixed integer program described in the previous section presents some unique challenges. Network design problems have been extensively studied in the literature (see, for example, 9] 10] 11] 12] [13], 15] 16] 18] and references therein) with good experimental results using cut and branch (see [14] for mathematical programming basics) However, traditional models tend to stress one particular feature, typically link transmission systems (modeled as edge capacities in an underlying network) ....

....analysis as was just mentioned may no longer yield a strong, or even valid, bound, but the LP based procedure remains valid. A critical step in generating inequalities (18) or (19) is the choice of the cutset C. Following the work of other researchers, and, in particular, our work in [12] and [13], we used an approach that combines some enumeration and also the choice of cutsets that are tight in the current LP relaxation. Another trick is to maintain a list or cache of cutsets that were used in previous inequalities to test them against the current LP relaxation. Yet another trick ....

D. Bienstock, S. Chopra, O. Gnlk,and C. Tsai, "Minimum cost capacity installation for multicommodity network flows," (1995) to appear, Mathematical Programming (Special issue on Computational Integer Programming).

....relaxations of formulations of several network design problems, and (b) pure multicommodity flow problems. Unless specifically stated, all experiments were carried out on a 667 Mhz ev5 Alpha computer, with 1GB of memory and 8MB of cache. 5.1 Network design problems. The basic problem is described [3]. Given a directed graph G and fractional multicommodity demands, the mixed integer program consists of installing integer capacities on the edges of the graph, at minimum cost, so that the demands can be routed. The linear programming relaxations, strengthened with facet defining inequalities, ....

D. Bienstock, S. Chopra, O. Gunluk and C. Tsai, Minimum Cost Capacity Installation for Multicommodity Network Flows, Math. Programming 81 (1998), 177-199.

....inequality is not a rounded metric inequality. It is a Chv atal Gomory cut obtained from partition inequalities. 3.2 Complete Description of P(C) We next show that partition inequalities (R) and the total capacity inequality (17) completely define P (C) for the complete graph on 3 nodes. See [3] for a proof of the following result. Proposition 3.2 For any complete graph with jV j = 3, the partition inequalities along with the total capacity inequality completely define P (C) Remark: The total capacity inequality can be extended to a valid inequality for graphs with more than 3 nodes ....

Bienstock, D., Chopra, S., Gunluk, O., and Tsai, C-Y, "Minimum Cost Capacity Installation for Multicommodity Network Flows," CORE Discussion Paper, Universit'e Catholique de Louvain, Belgium, (1995).

....[6] as it contains the set cover problem (and perhaps more to point, the fixed charge network design problem, and thus the Steiner tree problem) as a special case. The problem has been known by names such as network loading problem, 10] 12] minimum cost capacity installation problem, [4])and others. We will denote the particular version of the problem that we study by CEP in the sequel, and will define it below. Our primary motivation for studying CEP is that it naturally arises as part of a much larger and complex problem concerning ATM (asynchronous transfer mode) network ....

....that are generated algorithmically. A second class of models considered in [18] can in addition handle side constraints, such as survivability constraints. Even more recently, a similar projection approach has been implemented by Barahona (see [1] and Bienstock, Chopra, Gunluk and Tsai (see [4]) This approach may well be competitive with the multicommodity flows formulation. Several facet defining inequalities for the projection are described in [12] Next, we briefly introduce the notation used in this paper. In what follows, the set of all real numbers is denoted by R, and ....

[Article contains additional citation context not shown here]

D. Bienstock, S. Chopra, O. Gunluk and C.-Y. Tsai, Minimum-cost capacity installation for multicommodity network flows (January 1995), submitted.

....incorporate these additional reliability constraints. 4. Solution procedure and heuristics The mixed integer program described in the previous section presents some unique challenges. Network design problems have been extensively studied in the literature (see, for example, 9] 10] 11] 12] [13], 15] 16] 18] and references therein) with good experimental results using cut and branch (see [14] for mathematical programming basics) However, traditional models tend to stress one particular feature, typically link transmission systems (modeled as edge capacities in an underlying network) ....

....analysis as was just mentioned may no longer yield a strong, or even valid, bound, but the LP based procedure remains valid. A critical step in generating inequalities (18) or (19) is the choice of the cutset C. Following the work of other researchers, and, in particular, our work in [12] and [13], we used an approach that combines some enumeration and also the choice of cutsets that are tight in the current LP relaxation. Another trick is to maintain a list or cache of cutsets that were used in previous inequalities to test them against the current LP relaxation. Yet another trick ....

D. Bienstock, S. Chopra, O. Günlük,and C. Tsai, "Minimum cost capacity installation for multicommodity network flows," (1995) to appear, Mathematical Programming (Special issue on Computational Integer Programming).

No context found.

D. Bienstock, S. Chopra, O. Gunluk, C. Tsai, Minimum cost capacity installation for multicommodity network flows, Math. Programming B 81 2 (1998) 177--199.

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D. Bienstock, S. Chopra, O. Gunluk, and C-Y. Tsai. Minimum cost capacity installation for multicommodity ow networks. Mathematical Programming, 81:177-199, 1998. 7

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D. Bienstock, S. Chopra, O. Gunluk, and C.-Y. Tsai, "Minimum cost capacity installation for multicommodity network flows," Mathematical Programming, vol. 81, pp. 177--199, 1998.

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D. Bienstock, S. Chopra, O. Gunluk, and C.-Y. Tsai. Minimum cost capacity installation for multicommodity networks. Mathematical Programming, 81:177--199, 1998.

No context found.

Bienstock, D., S. Chopra, and O. Gunluk (1998), Minimum cost capacity installation for multicommodity network flows, Mathematical Programming, Series B, Vol. 81, No. 2-1, pp. 177-199.

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