Using Lagrangian dual information to generate degree constrained spanning trees

Andrade, Rafael; Lucena, Abílio; Maculan, Nelson

doi:10.1016/j.dam.2005.06.011

Cited by 29 publications

(43 citation statements)

References 21 publications

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“…In the current tree tr (see Figure 3a), the edge (8,10) is a preferred replacing edge, nca tr (s, 8) = 7, nca tr (s, 10) = 12, lownca tr ((8, 10), s) = 7, upnca tr ((8, 10), s) = 12, low tr ((8,-10), s) = 8 and up tr ((8, 10), s) = 10. The edges (7,11) and (11,12) are preferred replaceable edges of (8,10) because these edges belong to path tr (7,12). The path induced by tr is s, 3, 4, 6, 7, 11, 12, t .…”

Section: The Edge-replacement Based Neighborhoodmentioning

confidence: 99%

“…Lines 3-4 compute the set of connected components CC of the graph generated from the current graph by removing all edges E of paths of S i . For each graph G i of these connected components and each set of commodities T i that belong to G i , we perform recursively the LS-Recursive method (see lines [6][7][8].…”

Section: The Ls-r Algorithmmentioning

confidence: 99%

“…Most of these COT/COP problems are NP-hard. They are often approached by dedicated algorithms including exact methods, such as the Lagrangian-based heuristic [7], the ILP-based algorithm using directed cuts [25], the Lagrangian-based branch and bound in [15], and the vertex labeling algorithm from [30]; there are also meta-heuristic algorithms such as a hybrid evolutionary algorithm [19], ant colony optimization [21], and local search [20]. These techniques exploit the structure of the constraints and the objective functions but are often difficult to extend or reuse.…”

Section: Introductionmentioning

confidence: 99%

“…These problems consist of finding one or more trees (or paths) on a given graph satisfying some given constraints while minimizing or maximizing an objective function. Some COT problems have been considered and solved in the literature, e.g., Degree Constrained Minimum Spanning Tree (DCMST) [45,7], Bounded Diameter Minimum Spanning Tree (BDMST) [35], Capacitated Minimum Spanning Tree problem (CMST) [56,3], Minimum Diameter Spanning Tree (MDST) [50], EdgeWeighted k-Cardinality Tree (KCT), [20,25], Steiner Minimal Tree (SMT) [66,28], Optimum Communication Spanning Tree problems (OCST) [32], etc. We also see many COP problems which have been studied and solved in the literature.…”

Section: Introductionmentioning

confidence: 99%

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LS(Graph): a constraint-based local search for constraint optimization on trees and paths

2012

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Constrained optimum tree (COT) and constrained optimum path (COP) problems arise in many real-life applications and are ubiquitous in communication networks. They have been traditionally approached by dedicated algorithms, which are often hard to extend with side constraints and to apply widely. This paper proposes a constraint-based local search framework for COT/COP applications, bringing the compositionality, reuse, and extensibility at the core of constraint-based local search and constraint programming systems. The modeling contribution is the ability to express compositional models for various COT/COP applications at a high level of abstraction, while cleanly separating the model and the search procedure. The main technical contribution is a connected neighborhood based on rooted spanning trees to find high-quality solutions to COP problems. This framework is applied to some COT/COP problems, e.g., the quorumcast routing problem, the edge-disjoint paths problem, and the routing and wavelength assignment with delay side constraints problem. Computational results show the potential importance of the approach.

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Section: The Edge-replacement Based Neighborhoodmentioning

confidence: 99%

Section: The Ls-r Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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LS(Graph): a constraint-based local search for constraint optimization on trees and paths

2012

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“…Unlike MDC-MST, DC-MST is a well-studied problem. Some of the notable contributions on DC-MST are Deo and Hakimi [8], Savelsbergh and Volgenant [9], Zhou and Gen [10], Knowles and Corne [11], Caccetta and Hill [12], Ribeiro and Souza [13], Andrade et al [14], and Krishnamoorthy et al [15].…”

Section: Introductionmentioning

confidence: 99%

Min-degree constrained minimum spanning tree problem: New formulation via Miller–Tucker–Zemlin constraints

Akgün

Tansel

2010

Computers & Operations Research

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Given an undirected network with positive edge costs and a positive integer d > 2, the minimum-degree constrained minimum spanning tree problem is the problem of finding a spanning tree with minimum total cost such that each non-leaf node in the tree has a degree of at least d. This problem is new to the literature while the related problem with upper bound constraints on degrees is well studied. Mixedinteger programs proposed for either type of problem is composed, in general, of a tree-defining part and a degree-enforcing part. In our formulation of the minimum-degree constrained minimum spanning tree problem, the tree-defining part is based on the Miller-Tucker-Zemlin constraints while the only earlier paper available in the literature on this problem uses single and multi-commodity flow-based formulations that are well studied for the case of upper degree constraints. We propose a new set of constraints for the degree-enforcing part that lead to significantly better solution times than earlier approaches when used in conjunction with Miller-Tucker-Zemlin constraints.

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Lower and upper bounds for the degree‐constrained minimum spanning tree problem

Cunha

Lucena

2007

Networks

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In this paper, we propose a Lagrangian Non-Delayed Relax-and-Cut algorithm for the Degree-Constrained Minimum Spanning Tree Problem (DCMSTP). Since degreeconstrained trees are the common vertices of the Spanning Tree and the b-Matching polytopes, strengthened DCMSTP lower bounds are generated through the use of Blossom Inequalities (BIs). BIs are facet defining for the b-Matching Polytope and, to the best of our knowledge, have never been used before for DCMSTP. Lagrangian information is also used here, in a Kruskal style algorithm (followed by local search) to obtain valid DCMSTP upper bounds. Whenever optimality could not be proven by Relax-and-Cut alone, Lagrangian lower bounds are carried over to a cutting plane algorithm, without the need for solving any separation problem. This is accomplished after identifying, at Relax-and-Cut, a small set of attractive Subtour Elimination Constraints and BIs. Computational results, on Euclidean and on random cost DCMSTP instances, indicate that our Relax-and-Cut lower bounds either dominate or else are competitive with linear programming and Lagrangian relaxation lower bounds found in the literature. Furthermore, our upper bounds appear competitive with the best in the literature.

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Using Lagrangian dual information to generate degree constrained spanning trees

Cited by 29 publications

References 21 publications

LS(Graph): a constraint-based local search for constraint optimization on trees and paths

LS(Graph): a constraint-based local search for constraint optimization on trees and paths

Min-degree constrained minimum spanning tree problem: New formulation via Miller–Tucker–Zemlin constraints

Lower and upper bounds for the degree‐constrained minimum spanning tree problem

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