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.

Given a graph with edge weights satisfying the triangle inequality, and a degree bound for each vertex, the problem of computing a low weight spanning tree such that the degree of each vertex is at most its specified bound is considered. In particular, modifying a given spanning tree T using adoptions to meet the degree constraints is considered. A novel network-flow based algorithm for finding a good sequence of adoptions is introduced. The method yields a better performance guarantee than any previously obtained. Equally importantly, it takes this approach to the limit in the following sense: if any performance guarantee that is solely a function of the topology and edge weights of a given tree holds for any algorithm at all, then it also holds for our algorithm. The performance guarantee is the following. If the degree constraint d(v) for each v is at least 2, the algorithm is guaranteed to find a tree whose weight is at most the weight of the given tree times 2 \Gamma min n d(v)\Gamma2 deg T (v)\Gamma2 : deg T (v) ? 2 o ; where deg T (v) is the initial degree of v. Examples are provided in which no lighter tree meeting the degree constraint exists. Linear-time algorithms are provided with the same worst-case performance guarantee. Choosing T to be a minimum spanning tree yields approximation algorithms for the general problem on geometric graphs with distances induced by various Lp norms. Finally, examples of Euclidean graphs are provided in which the ratio of the lengths of an optimal Traveling Salesman path and a minimum spanning tree can be arbitrarily close to 2.

is a fundamental design choice in a genetic algorithm. This paper describes a novel coding of spanning trees in a genetic algorithm for the degree-constrained minimum spanning tree problem. For a connected, weighted graph, this problem seeks to identify the shortest spanning tree whose degree does not exceed an upper bound k 2. In the coding, chromosomes are strings of numerical weights associated with the target graph's vertices. The weights temporarily bias the graph's edge costs, and an extension of Prim's algorithm, applied to the biased costs, identifies the feasible spanning tree a chromosome represents. This decoding algorithm enforces the degree constraint, so that all chromosomes represent valid solutions and there is no need to discard, repair, or penalize invalid chromosomes. On a set of hard graphs whose unconstrained minimum spanning trees are of high degree, a genetic algorithm that uses this coding identifies degree-constrained minimum spanning trees that are on average shorter than those found by several competing algorithms.

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

. Given an undirected graph with weights associated with its edges, the degreeconstrained minimum spanning tree problem consists in finding a minimum spanning tree of the given graph, subject to constraints on node degrees. We propose a variable neighborhood search heuristic for the degree-constrained minimum spanning tree problem, based on a dynamic neighborhood model and using a variable neighborhood descent iterative improvement algorithm for local search. Computational experiments illustrating the effectiveness of the approach on benchmark problems are reported. Key words. Combinatorial optimization, degree-constrained minimum spanning tree, local search, metaheuristics, variable neighborhoods 1. Introduction. Let G = (V; E) be a connected undirected graph, where V is the set of nodes and E denotes the set of edges. Given a non-negative weight function w : E ! IR + associated with its edges and a non-negative integer valued degree function b : V ! IN associated with its nodes, th...

Finding the degree-constrained minimum spanning tree (d-MST) of a graph is a well studied NP-hard problem which is important in network design. We introduce a new method which improves on the best technique previously published for solving the d-MST, either using heuristic or evolutionary approaches. The basis of this encoding is a spanning-tree construction algorithm which we call the Randomised Primal Method (RPM), based on the well-known Prim's algorithm [6], and an extension [4] which we call `d-Prim's'. We describe a novel encoding for spanning trees, which involves using the RPM to interpret lists of potential edges to include in the growing tree. We also describe a random graph generator which produces particularly challenging d-MST problems. On these and other problems, we find that an evolutionary algorithm (EA) using the RPM encoding outperforms the previous best published technique from the operations research literature, and also outperforms simulated...

Finding the degree-constrained minimum spanning tree (d-MST) of a graph is a wellstudied NP-hard problem of importance in communications network design and other network-related problems. In this paper we describe some previously proposed algorithms for solving the problem, and then introduce a novel tree construction algorithm called the Randomised Primal Method (RPM) which builds degree-constrained trees of low cost from solution vectors taken as input. RPM is applied in three stochastic iterative search methods: simulated annealing, multi-start hillclimbing, and a genetic algorithm. While other researchers have mainly concentrated on finding spanning trees in Euclidean graphs, we consider the more general case of random graph problems. We describe two random graph generators which produce particularly challenging d-MST problems. On these and other problems we find that the genetic algorithm employing RPM outperforms simulated annealing and multi-start hillclimbing. Our experimental ...

Given an unorganized two-dimensional point cloud, we address the problem of efficiently constructing a single aesthetically pleasing closed interpolating shape, without requiring dense or uniform spacing. Using Gestalt’s laws of proximity, closure and good continuity as guidance for visual aesthetics, we require that our constructed shape be minimal perimeter, non-self intersecting and manifold. We find that this yields visually pleasing results. Our algorithm is distinct from earlier shape reconstruction approaches, in that it exploits the overlap between the desired shape and a related minimal graph, the Euclidean Minimum Spanning Tree (EMS T). Our algorithm segments the EMS T to retain as much of it as required and then locally partitions and solves the problem efficiently. Comparison with some of the best currently known solutions shows that our algorithm yields better results. Keywords:

Abstract. The degree-constrained minimum spanning tree (DCMST) is relevant in the design of networks. It consists of finding a spanning tree whose nodes do not exceed a given maximum degree and whose total edge length is minimum. We design a primal branch-and-cut algorithm that solves instances of the problem to optimality. Primal methods have not been used extensively in the past, and their performance often could not compete with their standard ‘dual ’ counterparts. We show that primal separation procedures yield good bounds for the DCMST problem. On several instances, the primal branch-and-cut program turns out to be competitive with other methods known in the literature. This shows the potential of the primal method. 1

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