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12
An Efficient Evolutionary Algorithm for the DegreeConstrained Minimum Spanning Tree Problem
, 2000
"... 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 degreeconstrained minimum spanning tree problem. For a weighted, undi ..."
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Cited by 26 (5 self)
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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 degreeconstrained 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 timecomplexity. 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.
A Weighted Coding in a Genetic Algorithm for the DegreeConstrained Minimum Spanning Tree Problem
, 2000
"... is a fundamental design choice in a genetic algorithm. This paper describes a novel coding of spanning trees in a genetic algorithm for the degreeconstrained minimum spanning tree problem. For a connected, weighted graph, this problem seeks to identify the shortest spanning tree whose degree does n ..."
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Cited by 20 (4 self)
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is a fundamental design choice in a genetic algorithm. This paper describes a novel coding of spanning trees in a genetic algorithm for the degreeconstrained 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 degreeconstrained minimum spanning trees that are on average shorter than those found by several competing algorithms.
EdgeSets: An Effective Evolutionary Coding of Spanning Trees
, 2002
"... 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 d ..."
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Cited by 18 (8 self)
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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
Evolutionary Local Search for the EdgeBiconnectivity Augmentation Problem
 INFORMATION PROCESSING LETTERS
, 2002
"... This paper considers the problem of augmenting a given graph by a cheapest possible set of additional edges in order to make the graph edgebiconnected. An application is the extension of an existing communication network to become robust against single linkfailures. An evolutionary algorithm (EA) ..."
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Cited by 10 (2 self)
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This paper considers the problem of augmenting a given graph by a cheapest possible set of additional edges in order to make the graph edgebiconnected. An application is the extension of an existing communication network to become robust against single linkfailures. An evolutionary algorithm (EA) is presented which includes an e#ective preprocessing of the problem data and a local improvement procedure that is applied during initialization, recombination, and mutation. In this way, the EA searches the space of feasible, locally optimal solutions only. The variation operators were designed with particular emphasis on low computational e#ort and strong locality. Empirical results indicate the superiority of the new approach over two previous heuristic methods
Identifying structural mechanisms in standard genetic programming, in Genetic and evolutionary computing
 GECCO 2003, E. Cantu  Paz et
, 2003
"... Abstract. This paper presents a hypothesis about an undiscovered class of mechanisms that exist in standard GP. Rather than being intentionally designed, these mechanisms would be an unintended consequence of using trees as information structures. A model is described that predicts outcomes in GP th ..."
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Cited by 9 (3 self)
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Abstract. This paper presents a hypothesis about an undiscovered class of mechanisms that exist in standard GP. Rather than being intentionally designed, these mechanisms would be an unintended consequence of using trees as information structures. A model is described that predicts outcomes in GP that would arise solely from such mechanisms. Comparisons with empirical results from GP lend support to the existence of these mechanisms. 1
WeightBiased EdgeCrossover in Evolutionary Algorithms for Two Graph Problems
 Proceedings of the 2001 ACM symposium on Applied computing
"... Many optimization problems on weighted graphs seek a subset of the graph's edges that has minimum weight and satisfies the problem's constraints. Two examples are the traveling salesman problem (TSP) and the degreeconstrained minimum spanning tree problem (dMSTP). Heuristics like evoluti ..."
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Cited by 7 (3 self)
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Many optimization problems on weighted graphs seek a subset of the graph's edges that has minimum weight and satisfies the problem's constraints. Two examples are the traveling salesman problem (TSP) and the degreeconstrained minimum spanning tree problem (dMSTP). Heuristics like evolutionary algorithms often construct candidate solutions to such problems iteratively, repeatedly including an edge selected from those currently eligible. Not surprisingly, low weight edges usually predominate in good and optimal solutions, an observation we confirm empirically for the TSP and the dMSTP. This suggests that any process that builds candidate solutions should, with higher probability, select edges of lower weight. We incorporate into crossover operators and compare in a genetic algorithm four edgeselection techniques: random, greedy, according to probabilities inversely proportional to the edges' weights, and 2tournament. Tests on instances of the TSP and the dMSTP indicate that with the weightbiased techniques, the GA identifies better solutions faster than with random edgeselection.
A Predecessor Coding in an Evolutionary Algorithm for the Capacitated Minimum Spanning Tree Problem
 Late Breaking Papers at the 2000 Genetic and Evolutionary Computation Conference, pages 309–316, Las Vegas, NV
"... 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. Initiali ..."
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Cited by 5 (3 self)
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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 lowcost links. Empirical results on a set of standard test problems indicate that the EA performs better than two other heuristic techniques.
Codings and operators in two genetic algorithms for the leafconstrained minimum spanning tree problem
 International Journal of Applied Mathematics and Computer Science
, 2004
"... The features of an evolutionary algorithm that most determine its performance are the coding by which its chromosomes represent candidate solutions to its target problem and the operators that act on that coding. Also, when a problem involves constraints, a coding that represents only valid solution ..."
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Cited by 4 (1 self)
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The features of an evolutionary algorithm that most determine its performance are the coding by which its chromosomes represent candidate solutions to its target problem and the operators that act on that coding. Also, when a problem involves constraints, a coding that represents only valid solutions and operators that preserve that validity represent a smaller search space and result in a more effective search. Two genetic algorithms for the leafconstrained minimum spanning tree problem illustrate these observations. Given a connected, weighted, undirected graph G with n vertices and a bound ℓ, this problem seeks a spanning tree on G with at least ℓ leaves and minimum weight among all such trees. A greedy heuristic for the problem begins with an unconstrained minimum spanning tree on G, then economically turns interior vertices into leaves until their number reaches ℓ. One genetic algorithm encodes candidate trees with Prüfer strings decoded via the Blob Code. The second GA uses strings of length n−ℓ that specify trees ’ interior vertices. Both GAs apply operators that generate only valid chromosomes. The latter represents and searches a much smaller space. In tests on 65 instances of the problem, both Euclidean and with weights chosen randomly, the BlobCoded GA cannot compete with the greedy heuristic, but the subsetcoded GA consistently identifies leafconstrained spanning trees of lower weight than the greedy heuristic does, particularly on the random instances.
Biased mutation operators for subgraphselection problems
 IN IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION
, 2004
"... Many graph problems seek subgraphs of minimum weight that satisfy a set of constraints. Examples include the minimum spanning tree problem (MSTP), the degreeconstrained minimum spanning tree problem (dMSTP), and the traveling salesman problem (TSP). Lowweight edges predominate in optimum solution ..."
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Cited by 4 (0 self)
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Many graph problems seek subgraphs of minimum weight that satisfy a set of constraints. Examples include the minimum spanning tree problem (MSTP), the degreeconstrained minimum spanning tree problem (dMSTP), and the traveling salesman problem (TSP). Lowweight edges predominate in optimum solutions to such problems, and the performance of evolutionary algorithms (EAs) is often improved by biasing variation operators to favor these edges. We investigate the impact of biased edgeexchange mutation. In a largescale empirical investigation, we study the distributions of edges in optimum solutions of the MSTP, the dMSTP, and the TSP in terms of the edges ’ weightbased ranks. We approximate these distributions by exponential functions and derive approximately optimal probabilities for selecting edges to be incorporated into candidate solutions during mutation. A theoretical analysis of the expected running time