VNS and second order heuristics for the min-degree constrained minimum spanning tree problem

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Abstract

Given an undirected graph with weights associated with its edges, the min-degree constrained minimum spanning tree (md-MST) problem consists in finding a minimum spanning tree of the given graph, imposing minimum degree constraints in all nodes except the leaves. This problem was recently proposed in Almeida et al. [Min-degree constrained minimum spanning tree problem: Complexity, proprieties and formulations. Operations Research Center, University of Lisbon, Working-paper no. 6; 2006], where its theoretical complexity was characterized and showed to be NP-hard.

The present paper discusses variable neighborhood search (VNS) metaheuristics addressing the md-MST. A so-called enhanced version of a second order (ESO) repetitive technique is also considered, in order to guide the search in both shaking and improvement phases of the VNS method. A Kruskal based greedy heuristic adapted to the md-MST is also presented, being used within the ESO framework. VNS randomized methodologies are also discussed. These are the first heuristics to the md-MST ever proposed in the literature. Computational experiments are conducted on instances adapted from benchmark ones used in the context of the well-known degree constraint minimum spanning tree problem. These experiments have shown that randomized VNS methods enclosing an ESO algorithm can produce very interesting results. In particular, that a simpler VNS randomized methodology might be taken into account when very high dimensional instances are under consideration.

Introduction

Let G=(V,E) be a complete undirected graph, where V={1,,n} is the set of nodes and E the set of edges. Consider that there are associated positive costs w(e) to each edge eE. Given a positive integer constant d, the Min-Degree Constrained Minimum Spanning Tree (md-MST) problem consists in finding a spanning tree T of G with minimum total edge cost and where each node iV either has degree at least d or is a leaf.

Fig. 1 gives an example of two feasible solutions to the md-MST problem, for a graph G with 10 nodes and considering d=3. In these solutions, all the internal nodes (in grey) have at least degree 3, while all the other nodes are leafs.

These solutions are no longer feasible if the degree bound is 4 (i.e., d=4). In the first tree all central nodes have degree less than 4, while in the second solution nodes 3 and 9 also violate this bound.

This problem is closely related with the well-known Degree Constrained Minimum Spanning Tree (d-MST or DCMST) problem, where also a minimum cost spanning tree T of G is seek, but with nodes degree at most d for all nodes of T.

Applications of the md-MST problem can be associated with a network where nodes are seldom requested in high flow amounts, while in the d-MST nodes are frequently used in small flow quantities. The d-MST problem has long been discussed in the literature, being found in many network design applications, namely in VLSI design, road systems, energy and computer communication networks, etc (see [17], [22], [18], [8]). In all these applications the d-MST seeks for over access prevention, while the md-MST looks for a minimal network having special nodes where information or facilities are centralized, instead of being spread out.

The md-MST also resembles the hub location problem. However, the hub location is a network flow problem, while the md-MST belongs to the pure network design family.

The present paper proposes the combination of two metaheuristics. It considers the well-known Variable Neighborhood Search (VNS) method, proposed by Mladenović and Hansen [16], and uses an enhanced second order (ESO) algorithm, proposed by Martins [15], within the improvement phase of the VNS framework, substituting the most common local search methodology. The ESO belongs to the family of repetitive techniques arising from the Second Order (SO) method proposed by Karnaugh [11] in 1976. The SO is based on repeated calls of a greedy heuristic. In each call a new constraint (not valid for the original problem) is introduced defining a sequence of subsets of the problem solution space. The SO stops when no further improvements are possible. Attempting for further improvements, the ESO allows the SO to continue in a deterministic manner, preventing against local optimal traps and cycling. When used in the VNS improvement phase, the ESO is called twice. First within the current kth neighborhood space and then within the whole solutions space.

This paper is organized as follows. Known results for the md-MST and proposed in [2] are presented in the next section. Then, in Section 3, we propose a greedy heuristic to the md-MST. SO based algorithms are described in Section 4, as well as VNS related methods, namely those involving the VNS and SO combinations. In Section 5, we report computational results from these heuristics, using known benchmark instances from the d-MST literature. Some of these instances have also been involved in the lower bounding tests reported in [2], being used to evaluate the quality of our bounds. Concluding remarks are given in the last section.

Section snippets

Previous results on the md-MST problem

The md-MST has been first introduced in Almeida et al. [2] and showed to be NP-hard for 4d<n/2. In fact, for d4, we can reduce the k-Dimensional Matching Problem, where k3, to the md-MST, using a constructive approach analogous to the proof of Partition into Triangles found in [7]. Complexity for d=3 is still open, while for d2 the problem becomes the MST.

Another result shows that for any feasible md-MST tree with S its set of central-nodes, then |S|(n-2)/(d-1), setting an upper bound

Greedy heuristic for the md-MST problem

One of the most popular techniques to build a minimum spanning tree on a graph is the Kruskal's algorithm, first proposed in [14]. It is known to find an optimal MST solution in polynomial time (see, e.g. [1]).

The Kruskal's algorithm is a one-pass procedure that builds a spanning tree from scratch by adding one edge at a time. A fast implementation can be obtained if the edges are previously sorted in nondecreasing order of their costs. Considering the edges in this order, we take one edge at a

VNS and second order combined algorithms

In this section we describe the well known VNS method and an Enhanced version of the SO algorithm. We propose a combined version involving the two mentioned metaheuristics.

Computational results

Computational results on the md-MST problem are presented in this section. For this purpose we used the ESO algorithm described in Section 4.2 and the various VNS/ESO and Rk-VNS/ESO combined algorithms described in Section 4.3. All the codes have been implemented in Fortran 90, using the Fortran PowerStation 4.0 compiler. The tests were ran on a Pentium IV 3.2 GHz with 512 Mbytes of RAM memory.

We conducted our computational experiments on instance classes adapted from those used as benchmark for

Conclusions

The md-MST problem has been recently proposed in [2], to which complexity, proprieties and models have been discussed.

Heuristic methods to this problem are first proposed in the present paper. These include a greedy heuristic, based on Kruskal's algorithm, the SO method proposed in [11] and an enhanced version of this method (designated by ESO) described in [15]. Then, four combined methodologies have been considered. The first combines the ESO with the well known VNS. The second and the third

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