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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 24 (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 18 (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.
A New Evolutionary Approach to the Degree Constrained Minimum Spanning Tree Problem
 IEEE Transactions on Evolutionary Computation
, 2000
"... Finding the degreeconstrained minimum spanning tree (dMST) of a graph is a well studied NPhard problem which is important in network design. We introduce a new method which improves on the best technique previously published for solving the dMST, either using heuristic or evolutionary app ..."
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Cited by 11 (0 self)
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Finding the degreeconstrained minimum spanning tree (dMST) of a graph is a well studied NPhard problem which is important in network design. We introduce a new method which improves on the best technique previously published for solving the dMST, either using heuristic or evolutionary approaches. The basis of this encoding is a spanningtree construction algorithm which we call the Randomised Primal Method (RPM), based on the wellknown Prim's algorithm [6], and an extension [4] which we call `dPrim'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 dMST 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...
A New Evolutionary Approach to the DegreeConstrained Minimum Spanning Tree Problem
 IEEE Transactions on Evolutionary Computation
, 1999
"... Finding the degreeconstrained minimum spanning tree (dMST) of a graph is a wellstudied NPhard problem of importance in communications network design and other networkrelated problems. In this paper we describe some previously proposed algorithms for solving the problem, and then introduce a nove ..."
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Cited by 10 (2 self)
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Finding the degreeconstrained minimum spanning tree (dMST) of a graph is a wellstudied NPhard problem of importance in communications network design and other networkrelated 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 degreeconstrained trees of low cost from solution vectors taken as input. RPM is applied in three stochastic iterative search methods: simulated annealing, multistart 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 dMST problems. On these and other problems we find that the genetic algorithm employing RPM outperforms simulated annealing and multistart hillclimbing. Our experimental ...
A Comparison of Encodings and Algorithms for Multiobjective Minimum Spanning Tree Problems
 In Proceedings of the 2001 Congress on Evolutionary Computation (CEC'01
, 1997
"... this paper we apply (appropriately modified) the best of recent methods for the (degreeconstrained) 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, inv ..."
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Cited by 8 (1 self)
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this paper we apply (appropriately modified) the best of recent methods for the (degreeconstrained) 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 populationbased 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.
Multicriteria network design using evolutionary algorithm
 Proc. Genetic and Evolutionary Computations Conference (GECCO), Lecture Notes in Computer Sciences
, 2003
"... Abstract. In this paper, we revisit a general class of multicriteria multiconstrained network design problems and attempt to solve, in a novel way, with Evolutionary Algorithms (EAs). A major challenge to solving such problems is to capture possibly all the (representative) equivalent and diverse ..."
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Cited by 6 (3 self)
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Abstract. In this paper, we revisit a general class of multicriteria multiconstrained network design problems and attempt to solve, in a novel way, with Evolutionary Algorithms (EAs). A major challenge to solving such problems is to capture possibly all the (representative) equivalent and diverse solutions. In this work, we formulate, without loss of generality, a bicriteria bi constrained communication network topological design problem. Two of the primary objectives to be optimized are network delay and cost subject to satisfaction of reliability and flowconstraints. This is a NPhard problem so we use a hybrid approach (for initialization of the population) along with EA. Furthermore, the twoobjective optimal solution front is not known a priori. Therefore, we use a multiobjective EA which produces diverse solution space and monitors convergence; the EA has been demonstrated to work effectively across complex problems of unknown nature. We tested this approach for designing networks of different sizes and found that the approach scales well with larger networks. Results thus obtained are compared with those obtained by two traditional approaches namely, the exhaustive search and branch exchange heuristics. 1
Multiobjective EA approach for improved quality of solutions for spanning tree problem
 in: Proc. Internat. Conf. Evolutionary MultiCriterion Optimization (EMO), Lecture Notes in Computer Science
, 2005
"... Abstract. The problem of computing spanning trees along with specific constraints is mostly NPhard. Many approximation and stochastic algorithms which yield a single solution, have been proposed. In this paper, we formulate the generic multiobjective spanning tree (MOST) problem and consider edge ..."
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Cited by 5 (0 self)
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Abstract. The problem of computing spanning trees along with specific constraints is mostly NPhard. Many approximation and stochastic algorithms which yield a single solution, have been proposed. In this paper, we formulate the generic multiobjective spanning tree (MOST) problem and consider edgecost and diameter as the two objectives. Since the problem is hard, and the Paretofront is unknown, the main issue in such probleminstances is how to assess the convergence. We use a multiobjective evolutionary algorithm (MOEA) that produces diverse solutions without needing a priori knowledge of the solution space, and generate solutions from multiple tribes in order to assess movement of the solution front. Since no experimental results are available for MOST, we consider three well known diameterconstrained minimum spanning tree (dcMST) algorithms including randomized greedy heuristics (RGH) which represents the current state of the art on the dcMST, and modify them to yield a (near) optimal solutionfronts. We quantify the obtained solution fronts for comparison. We observe that MOEA provides superior solutions in the entirerange of the Paretofront, which none of the existing algorithms could individually do. 1
An AntBased Algorithm for Finding DegreeConstrained Minimum Spanning Tree
"... A spanning tree of a graph such that each vertex in the tree has degree at most d is called a degreeconstrained spanning tree. The problem of finding the degreeconstrained spanning tree of minimum cost in an edge weighted graph is well known to be NPhard. In this paper we give an AntBased algori ..."
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Cited by 3 (0 self)
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A spanning tree of a graph such that each vertex in the tree has degree at most d is called a degreeconstrained spanning tree. The problem of finding the degreeconstrained spanning tree of minimum cost in an edge weighted graph is well known to be NPhard. In this paper we give an AntBased algorithm for finding low cost degreeconstrained spanning trees. Ants are used to identify a set of candidate edges from which a degreeconstrained spanning tree can be constructed. Extensive experimental results show that the algorithm performs very well against other algorithms on a set of 572 problem instances.
Benchmark Problem Generators and Results for the Multiobjective DegreeConstrained Minimum Spanning Tree Problem
 In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO2001
, 1997
"... Finding a minimumweight spanning tree ..."
A Parallel Algorithm for the DegreeConstrained Minimum Spanning Tree Problem Using Nearest Neighbor Chains
 In Proceedings of the Fourth International Symposium on Parallel Architectures, Algorithms, and Networks
, 1998
"... The DegreeConstrained Minimum Spanning Tree (dMST) problem attempts to find a minimum spanning tree with an added constraint that no nodes in the tree have a degree larger than a specified integer d. It is known that computing the dMST is NPhard for every d in the range 2 ≤ d ≤ (n − 2), where n ..."
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Cited by 2 (0 self)
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The DegreeConstrained Minimum Spanning Tree (dMST) problem attempts to find a minimum spanning tree with an added constraint that no nodes in the tree have a degree larger than a specified integer d. It is known that computing the dMST is NPhard for every d in the range 2 ≤ d ≤ (n − 2), where n denotes the total number of nodes. Several approximate algorithms (heuristics) have been proposed in the literature. We have previously proposed three approximate algorithms, IR, TCRNN, and TCNNC, for solving the dMST problem, the last two (TCRNN and TCNNC) take advantages of nearest neighbors and their properties. Our experimental results showed that both the TCRNN and TCNNC algorithms consistently produce spanning trees with a smaller weight (better qualityofsolution) than that of IR, but using slightly longer execution time. In this paper, we propose a new heap traversal technique that further improves the time efficiency of TCRNN and TCNNC. Our experiments using randomly generated, weighted graphs as inputs show that the TCNNC algorithm outperforms the other two approximate algorithms in terms of the execution time and qualityofsolution.