A branch and cut method for the degree-constrained minimum spanning tree problem

Caccetta, Louis; Hill, Stephen P.

doi:10.1002/1097-0037(200103)37:2<74::aid-net2>3.0.co;2-e

Cited by 34 publications

(20 citation statements)

References 14 publications

(36 reference statements)

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“…However, the result of MST-based clustering algorithm is easy to be affected by noise points. In other words, the existence of noise points may reduce the clustering effect of MST-based clustering algorithm, such as algorithms that proposed by paper [25], [28], [29], [32]. In order to avoid the influence of noise points, paper [26] propose a new MST-based algorithm (MST-LOF).…”

Section: Related Work a Mst-based And Center-based Clustering Almentioning

confidence: 99%

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A Novel Hybrid Clustering Algorithm Based on Minimum Spanning Tree of Natural Core Points

Huang

Cheng

et al. 2019

IEEE Access

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Clustering analysis has been widely used in pattern recognition, image processing, machine learning, and so on. It is a great challenge for most existing clustering algorithms to discover clusters with complex manifolds or great density variation. Most of the existing clustering needs manually set neighborhood parameter K to search the neighbor of each object. In this paper, we use natural neighbor to adaptively get the value of K and natural density of each object. Then, we define two novel concepts, natural core point and the distance between clusters to solve the complex manifold problem. On the basis of above-proposed concept, we propose a novel hybrid clustering algorithm that only needs one parameter M (the number of final clusters) based on minimum spanning tree of natural core points, called NCP. The experimental results on the synthetic dataset and real dataset show that the proposed algorithm is competitive with the state-of-the-art methods when discovering with the complex manifold or great density variation. INDEX TERMS Clustering, complex manifold, natural neighbor, natural core points, minimum spanning tree.

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Section: Related Work a Mst-based And Center-based Clustering Almentioning

confidence: 99%

“…MST-LOF employs LOF [27] to discard noise points whose density factors are larger than a threshold during the construction of MST. Many other MST-based clustering algorithms that maximize or minimize the degrees of link of the vertices are proposed in paper [28], [29].…”

Section: Introductionmentioning

confidence: 99%

A Novel Hybrid Clustering Algorithm Based on Minimum Spanning Tree of Natural Core Points

Huang

Cheng

et al. 2019

IEEE Access

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“…Even though this can be improved to n − 2 for non-root nodes, we retain n − 1 in (9) to be consistent with the form used by AMS. Constraints (10)- (12) give the appropriate set restrictions and non-negativity on the decision variables.…”

Section: Scf/def1mentioning

confidence: 99%

“…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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“…Consequently, our main contributions in this paper can be highlighted as follows. First, we propose three mixed-integer linear programming (MILP) models for the DCkMST problem and derive for each one an equivalent formulation using the handshaking lemma [21][22][23]. More precisely, we propose four compact polynomial formulations and two exponential models.…”

Section: Introductionmentioning

confidence: 99%

Degree-Constrained $k$ -Minimum Spanning Tree Problem

Adasme

Firoozabadi

2020

Complexity

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Let G V , E be a simple undirected complete graph with vertex and edge sets V and E , respectively. In this paper, we consider the degree-constrained k -minimum spanning tree (DC k MST) problem which consists of finding a minimum cost subtree of G formed with at least k vertices of V where the degree of each vertex is less than or equal to an integer value d ≤ k − 2 . In particular, in this paper, we consider degree values of d ∈ 2,3 . Notice that DC k MST generalizes both the classical degree-constrained and k -minimum spanning tree problems simultaneously. In particular, when d = 2 , it reduces to a k -Hamiltonian path problem. Application domains where DC k MST can be adapted or directly utilized include backbone network structures in telecommunications, facility location, and transportation networks, to name a few. It is easy to see from the literature that the DC k MST problem has not been studied in depth so far. Thus, our main contributions in this paper can be highlighted as follows. We propose three mixed-integer linear programming (MILP) models for the DC k MST problem and derive for each one an equivalent counterpart by using the handshaking lemma. Then, we further propose ant colony optimization (ACO) and variable neighborhood search (VNS) algorithms. Each proposed ACO and VNS method is also compared with another variant of it which is obtained while embedding a Q-learning strategy. We also propose a pure Q-learning algorithm that is competitive with the ACO ones. Finally, we conduct substantial numerical experiments using benchmark input graph instances from TSPLIB and randomly generated ones with uniform and Euclidean distance costs with up to 400 nodes. Our numerical results indicate that the proposed models and algorithms allow obtaining optimal and near-optimal solutions, respectively. Moreover, we report better solutions than CPLEX for the large-size instances. Ultimately, the empirical evidence shows that the proposed Q-learning strategies can bring considerable improvements.

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A branch and cut method for the degree-constrained minimum spanning tree problem

Cited by 34 publications

References 14 publications

A Novel Hybrid Clustering Algorithm Based on Minimum Spanning Tree of Natural Core Points

A Novel Hybrid Clustering Algorithm Based on Minimum Spanning Tree of Natural Core Points

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

Degree-Constrained $k$ -Minimum Spanning Tree Problem

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A branch and cut method for the degree-constrained minimum spanning tree problem

Cited by 34 publications

References 14 publications

A Novel Hybrid Clustering Algorithm Based on Minimum Spanning Tree of Natural Core Points

A Novel Hybrid Clustering Algorithm Based on Minimum Spanning Tree of Natural Core Points

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

Degree-Constrained k -Minimum Spanning Tree Problem

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Degree-Constrained $k$ -Minimum Spanning Tree Problem