VNS and second order heuristics for the min-degree constrained minimum spanning tree problem
Introduction
Let be a complete undirected graph, where is the set of nodes and the set of edges. Consider that there are associated positive costs to each edge . Given a positive integer constant , the Min-Degree Constrained Minimum Spanning Tree (-MST) problem consists in finding a spanning tree of with minimum total edge cost and where each node either has degree at least or is a leaf.
Fig. 1 gives an example of two feasible solutions to the -MST problem, for a graph with nodes and considering . 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., ). 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 (-MST or DCMST) problem, where also a minimum cost spanning tree of is seek, but with nodes degree at most for all nodes of .
Applications of the -MST problem can be associated with a network where nodes are seldom requested in high flow amounts, while in the -MST nodes are frequently used in small flow quantities. The -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 -MST seeks for over access prevention, while the -MST looks for a minimal network having special nodes where information or facilities are centralized, instead of being spread out.
The -MST also resembles the hub location problem. However, the hub location is a network flow problem, while the -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 th neighborhood space and then within the whole solutions space.
This paper is organized as follows. Known results for the -MST and proposed in [2] are presented in the next section. Then, in Section 3, we propose a greedy heuristic to the -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 -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 -MST problem
The -MST has been first introduced in Almeida et al. [2] and showed to be -hard for . In fact, for , we can reduce the -Dimensional Matching Problem, where , to the -MST, using a constructive approach analogous to the proof of Partition into Triangles found in [7]. Complexity for is still open, while for the problem becomes the MST.
Another result shows that for any feasible -MST tree with its set of central-nodes, then , setting an upper bound
Greedy heuristic for the -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 -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 R-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 -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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2017, European Journal of Operational ResearchCitation Excerpt :The reported times are average run times over 64 runs for each of the GA versions, evolving over 3000 generations with a population size of 3000. The computer used has an Intel Q9550 processor (Core2 2.83 gigahertz quadruple core) and 4 gigabytes RAM, so slightly slower than the systems used in the works used for comparison (Martinez & Cunha, 2014; Martins & de Souza, 2009; Murthy & Singh, 2013). The RAM capacity was of no consequence because the instance’s dimensions were rather small and we never needed to use more than a small amount of the overall capacity.
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2014, Discrete Applied MathematicsCitation Excerpt :As a direct result of the weaker bounds implied by MTZ, BB–MTZ usually needs to investigate many more nodes than BC. In order to compare the upper bounding capabilities of BC with the best VNS heuristics in [21], we also retrieve the best upper bound found by BC (named BUB tl) within the time limit imposed on the execution of the heuristics, defined in [21]. It is worth mentioning that all computational experiments reported in [21] were conducted on a Pentium IV machine, running at 3.2 GHz, with 512 MB of RAM memory.