A dual ascent approach for steiner tree problems on a directed graph

Wong, Richard T.

doi:10.1007/bf02612335

Cited by 310 publications

(193 citation statements)

References 11 publications

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“…Theorem 2: All vectors x and y satisfying (1)(2)(3)(4) and (6)(7)(8)(9)(10)(11) are associated with elementary cycles of G.…”

Section: Type Of Variables (Elementary Cycles) Numbermentioning

confidence: 99%

“…All combinatorial optimization problems that will be treated in this paper are associated with connected subgraphs of G. In the section 2 we present a nonsimultaneous flow formulation based on Claus & Maculan (1983), Beasley (1994), Wong (1984), Guyard (1985), Maculan (1986), Maculan (1987) and Maculan, Arpin & Nguyen (1988) for the Steiner tree problem in graphs and in Claus (1984) and Figueiredo & Maculan (2000) for the Travelling Salesman Problem (TSP) to guarantee that a subgraph has to be connected. We characterize all elementary cycles in a graph in section 3, and a particular model is developed for Hamiltonian cycles.…”

Section: Introduction and Definitionsmentioning

confidence: 99%

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Integer linear models with a polynomial number of variables and constraints for some classical combinatorial optimization problems

2003

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“…Theorem 2: All vectors x and y satisfying (1)(2)(3)(4) and (6)(7)(8)(9)(10)(11) are associated with elementary cycles of G.…”

Section: Type Of Variables (Elementary Cycles) Numbermentioning

confidence: 99%

Section: Introduction and Definitionsmentioning

confidence: 99%

Integer linear models with a polynomial number of variables and constraints for some classical combinatorial optimization problems

2003

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“…The vast literature on the Steiner network problem (see Winter [1987] for a recent survey) addresses issues of model formulation and polyhedral representations (e.g., Prodon, Liebling and Gr6flin [1985], Chopra and Rao [1988a] [1988b]), worst-case analysis of heuristics (e.g., Takahashi and Matsuyama [1980], Kou, Markowsky, and Berman [1981], Goemans and Bertsimas [1990]), and computational testing of optimization-based solution methods (e.g., Wong [1984], Beasley [1984], [1989]). The literature on the HND problem is relatively recent.…”

Section: Previous Researchmentioning

confidence: 99%

Modeling and Heuristic Worst-Case Performance Analysis of the Two-Level Network Design Problem

1994

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This paper studies a multi-facility network synthesis problem, called the Two-level Network Design (TLND) problem, that arises in the topological design of hierarchical communication, transportation, and electric power distribution networks. The nodes of a multi-level network have varying levels of importance; more critical or higher level nodes require more expensive higher grade interconnections. Given an undirected network with L possible facility types for each edge, and a partition of the nodes into L levels, the multilevel network design problem seeks a connected design that minimizes total cost while spanning all the nodes, and connecting nodes at each level via facilities of the corresponding or higher type. The TLND problem is a special case of multi-level network design with L = 2. This problem generalizes the well-known Steiner network problem and the hierarchical network design problem. In this paper, we study the relationship between alternative model formulations for this problem, and analyze the worst-case performance for a composite TLND heuristic based upon Steiner and spanning tree computations. When the ratio of higher to lower grade facility costs is the same for all edges, the worstcase performance ratio of the TLND heuristic is 4/3 if we can solve an embedded Steiner network problem optimally. For other cases, we express the TLND heuristic worst-case ratio in terms of the performance ratio of the Steiner solution method. A companion paper develops and tests a dual ascent procedure that generates tight upper and lower bounds on the optimal value of the multi-level network design problem

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“…However, researchers have studied several important special cases including the Steiner network problem and the HND problem. For the Steiner network problem, the -10-literature contains various optimal and optimization-based heuristic approaches (see Winter [1987] for a recent survey of Steiner network algorithms), including dynamic programming (Dreyfus and Wagner [1972]), Lagrangian relaxation (Beasley [1984], [1989]), dual ascent (Wong [1984]), polyhedral methods (Chopra et al [1990]), and simulated annealing (Osborne and Gillett [1991]). Chopra et al [19901 have solved to optimality Steiner network problems defined over complete networks containing up to 500 nodes, and sparse networks containing up to 2500 nodes, and 62500 edges.…”

Section: Dual-based Algorithm For the Tlnd Problemmentioning

confidence: 99%

“…Our proposed solution method for the MLND problem builds on the heritage of successful dual ascent procedures that researchers have previously developed for three related design problems-the Steiner network problem, the uncapacitated plant location problem, and the uncapacitated network design problems. Wong [1984] developed an efficient dual ascent algorithm to generate heuristic solutions (with associated performance guarantees) that were remarkably close to optimal. Erlenkotter [19781 and Balakrishnan, Magnanti, and Wong [1989] reported similar success using dual ascent algorithms for the uncapacitated plant location and (single-level) network design problems.…”

Section: Introductionmentioning

confidence: 99%

A Dual-Based Algorithm for Multi-Level Network Design

1994

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Given an undirected network with L possible facility types for each edge, and a partition of the nodes into L levels or grades, the Multi-level Network Design (MLND) problem seeks a fixed cost minimizing design that spans all the nodes and connects the nodes at each level by facilities of the corresponding or higher grade. This problem generalizes the well-known Steiner network problem and the hierarchical network design problem, and has applications in telecommunication, transportation, and electric power distribution network design. In a companion paper we studied alternative model formulations for a two-level version of this problem, and analyzed the worst-case performance of several heuristics based on Steiner network and spanning tree solutions. This paper develops a dual-based algorithm for the MLND problem. The method first performs problem preprocessing to fix certain design variables, and then applies a dual ascent procedure to generate upper and lower bounds on the optimal value. We report extensive computational results on large, random two-level test problems (containing up to 500 nodes, and 5,000 edges) with varying cost structures. The integer programming formulation of the largest of these problems has 20,000 integer variables and over 5 million constraints. Our tests indicate that the dual-based algorithm is very effective, producing solutions that are within 0.9% of optimality.network design, integer programming, dual ascent algorithm

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A dual ascent approach for steiner tree problems on a directed graph

Cited by 310 publications

References 11 publications

Integer linear models with a polynomial number of variables and constraints for some classical combinatorial optimization problems

Integer linear models with a polynomial number of variables and constraints for some classical combinatorial optimization problems

Modeling and Heuristic Worst-Case Performance Analysis of the Two-Level Network Design Problem

A Dual-Based Algorithm for Multi-Level Network Design

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