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The representation of candidate solutions and the variation operators are fundamental design choices in an evolutionary algorithm (EA). This paper proposes a novel representation technique and suitable variation operators for the degree-constrained minimum spanning tree problem. For a weighted, undirected graph G(V, E), this problem seeks to identify the shortest spanning tree whose node degrees do not exceed an upper bound d 2. Within the EA, a candidate spanning tree is simply represented by its set of edges. Special initialization, crossover, and mutation operators are used to generate new, always feasible candidate solutions. In contrast to previous spanning tree representations, the proposed approach provides substantially higher locality and is nevertheless computationally efficient; an offspring is always created in O(|V time. In addition, it is shown how problemdependent heuristics can be effectively incorporated into the initialization, crossover, and mutation operators without increasing the time-complexity. Empirical results are presented for hard problem instances with up to 500 vertices. Usually, the new approach identifies solutions superior to those of several other optimization methods within few seconds. The basic ideas of this EA are also applicable to other network optimization tasks.

This paper investigates how the use of redundant representations influences the performance of genetic and evolutionary algorithms. Representations are redundant if the number of genotypes exceeds the number of phenotypes. A distinction is made between synonymously and nonsynonymously redundant representations. Representation are synonymously redundant if the genotypes that represent the same phenotype are very similar to each other. Non-synonymously redundant representations do not allow genetic operators to work properly and result in a lower performance of evolutionary search. When using synonymously redundant representations, the performance of selectorecombinative genetic algorithms (GAs) depends on the modification of the initial supply. Theoretical models are developed that show the necessary population size to solve a problem and the number of generations goes with O(2 /r), where k r is the order of redundancy and r is the number of genotypic building blocks (BB) that represent the optimal phenotypic BB. Therefore, uniformly redundant representations do not change the behavior of GAs. Only by increasing r, which means overrepresenting the optimal solution, does GA performance increase. Therefore, non-uniformly redundant representations can only be used advantageously if a-priori information exists regarding the optimal solution. The validity of the proposed theoretical concepts is illustrated for the binary trivial voting mapping and the realvalued link-biased encoding. The empirical investigations show that the developed population sizing and time to convergence models allow an accurate prediction of the empirical results.

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The fundamental design choices in an evolutionary algorithm are its representation of candidate solutions and the operators that will act on that representation. We propose representing spanning trees in evolutionary algorithms for network design problems directly as sets of their edges, and we describe initialization, recombination, and mutation operators for this representation. The operators offer

this paper we apply (appropriately modified) the best of recent methods for the (degree-constrained) single objective MST problem to the multiobjective MST problem, and compare with a method based on Zhou and Gen's approach. Our evolutionary computation approaches, using the different encodings, involve a new population-based variant of Knowles and Corne's PAES algorithm. We find the direct encoding to considerably outperform the Prufer encoding. And we find that a simple iterated approach, based on Prim's algorithm modified for the multiobjective MST, also significantly outperforms the Prufer encoding.

When using genetic and evolutionary algorithms (GEAs) for the optimal communication spanning tree problem, the design of a suitable tree network encoding is crucial for finding good solutions. The link and node biased (LNB) encoding represents the structure of a tree network using a weighted vector and allows the GEA to distinguish between the importance of the nodes and links in the network. This paper investigates whether the encoding is unbiased in the sense that all trees are equally represented, and how the parameters of the encoding influence the bias. If the optimal solution is underrepresented in the population, a reduction in the GEA performance is unavoidable. The investigation reveals that the commonly used simpler version of the encoding is biased towards star networks, and that the initial population is dominated by only a few individuals. The more costly link-and-node-biased encoding uses not only a node-specific bias, but also a link-specific bias. Similarly to the node-biased encoding, the link-and-node-biased encoding is also biased towards star networks, especially when using a low weighting for the link-specific bias. The results show that by increasing the link-specific bias, that the overall bias of the encoding is reduced. If researchers want to use the LNB encoding, and they are interested in having an unbiased representation, they should use higher values for the weight of the link-specific bias. Nevertheless, they should also be aware of the limitations of the LNB encoding when using it for encoding tree problems. The encoding could be a good choice for the optimal communication spanning tree problem as the optimal solutions tend to be more star-like. However, for general tree problems the encoding should be used carefully.

This article presents an evolutionary algorithm (EA) for the capacitated minimum spanning tree problem occurring in telecommunication applications. The EA encodes a solution by a predecessor vector indicating for each node the preceding node at the path to the given central root node. Initialization, crossover, and mutation operators were specifically designed to provide strong locality and to enable an e#ective search in the space of feasible solutions only. Furthermore, local heuristics are applied to promote the inclusion of low-cost links. Empirical results on a set of standard test problems indicate that the EA performs better than two other heuristic techniques.

This paper presents an investigation into the properties of the optimal communication spanning tree (OCST) problem. The OCST problem finds a spanning tree that connects all nodes and satisfies their communication requirements for a minimum total cost. The paper compares the properties of randomly created solutions to the best solutions that are found using an evolutionary algorithm framework. The results show that on average the distance between the optimal solution and the minimum spanning tree (MST) that is calculated according to the distance weights is significantly smaller than the distance between a randomly created solution and the MST. This means, optimal solutions for the OCST problem are biased towards the MST defined on the distance weights alone. Consequently, the performance of optimization methods for the OCST problem can be increased if the search is biased towards MST-like solutions.

Abstract. To create a multicast routing distribution tree ensuring quality constraints is a NP-Hard problem. Many algorithm have been made to solve this issue, and genetic algorithms (GA) have shown better response times and solutions for this problem. This paper aligns related work by presenting algorithms in multicast routing using GAs and, taking its advantage, gives a proposal for solving this problem using a load balancing technique called multitree, modelled in previous work with the MHDB-S model.