## A New Evolutionary Approach to the Degree-Constrained Minimum Spanning Tree Problem (1999)

Venue: | IEEE Transactions on Evolutionary Computation |

Citations: | 10 - 2 self |

### BibTeX

@ARTICLE{Knowles99anew,

author = {Joshua Knowles and David Corne},

title = {A New Evolutionary Approach to the Degree-Constrained Minimum Spanning Tree Problem},

journal = {IEEE Transactions on Evolutionary Computation},

year = {1999},

volume = {4},

pages = {2000}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 ...

### Citations

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Citation Context ...Gamma 1, is NP-hard. This is obvious if one considers the special case where d = 2 because the dMST then becomes equivalent to the minimum-weight Hamiltonian path problem which is known to be NP-hard =-=[12]-=-. 2 The d-MST problem can be divided into two categories; Euclidean problems and random table problems. If the graph in which a tree is sought models a network laid out in a Euclidean space, such that... |

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Citation Context ...ch. This method, which we call RPM (Randomised Primal Method) is employed in three stochastic, iterative search techniques: multi-start hillclimbing, simulated annealing [10], and a genetic algorithm =-=[11]-=-. The quality of solutions found by these techniques are compared with each other, and also with those achieved by the dual simplex algorithm of [8] and the upper bound generated by Narula and Ho's d-... |

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Citation Context ...rithms acts on about 1% of genes and works by setting the gene to a new value from the same distribution as in the initialisation phase. In the genetic algorithm a standard uniform crossover operator =-=[15]-=- was also applied. 4 Different Techniques for Generating Random Graphs Random graphs are those in which the weight of each edge has been generated randomly from a uniform distribution within some pred... |

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Citation Context ...e number of evaluations is 10,000. A steady state genetic algorithm was used [11] with one crossover and one mutation per generation. A local mating selection was used following Collins and Jefferson =-=[16]-=-: The chromosomes are mapped onto a two-dimensional grid so that each has a location. 22 Only chromosomes that are near to each other on the grid can mate. This helps prevent the population from conve... |

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Citation Context ...arily obey the triangle inequality. In general, 2-d Euclidean problems are easier to solve than random table problems. This is because for every set of points in the plane there exists a degree-5 MST =-=[13]-=- whereas in random table problems the maximum degree of a vertex is, in theory, bounded only by the number of vertices in the graph. It is thus much more difficult to move from a solution of the uncon... |

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Citation Context ... polynomial time algorithm. In fact, for a given rational number Rs1, finding a spanning tree of maximum degree at most d, and of total weight at most R times that of the optimal solution, is NP-hard =-=[2]-=-. Several different approaches for solving the d-MST problem have been taken in previous research. Narula and Ho [3] investigated three methods, one of which (a branch and bound algorithm) is guarante... |

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Citation Context ...solve standard instances rising from a mean of 0.2 seconds (SD = 0.7s) for n = 30 to a mean of 74.2 seconds for n = 70 (SD = 369.5s), where n is the number of vertices 1 in the graphs. Khuller et al. =-=[6]-=- show that for an arbitrary collection of points in the plane there exists a degree-3 spanning tree of at most 1.5 times the MST and a degree-4 spanning tree of at most 1.25 times the MST. They also g... |

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Citation Context ...ost d, and of total weight at most R times that of the optimal solution, is NP-hard [2]. Several different approaches for solving the d-MST problem have been taken in previous research. Narula and Ho =-=[3]-=- investigated three methods, one of which (a branch and bound algorithm) is guaranteed to converge to a globally optimal solution. In their methods good upper bounds are generated by a modification to... |

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Citation Context ...ionary algorithm. 1 Introduction The minimum spanning tree (MST) of a graph is an important concept in the design of communication networks. It can be solved in polynomial time, and Moret and Shapiro =-=[1]-=- assess the practical performance of several constructive algorithms for solving it. In real networks, however, the vertices (or nodes) are usually subject to a degree-constraint. For example, many ex... |

20 | A network-flow technique for finding low-weight bounded-degree spanning trees
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Citation Context ... the MST. They also give algorithms that compute these trees in O(n) time, given an MST as part of the input. These results are also extended to Euclidean points in n-dimensional space. Fekete et al. =-=[7]-=- report a network flow technique that improves upon these results, working again in linear time. They do not consider the more difficult non-Euclidean case, however. Recent research by Boldon et al. [... |

18 |
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Citation Context ...the following, we will refer to the dual simplex method of Boldon et al. [8], using BF2 as the blacklisting function, as simply `BF2'. 3.4 Prufer Numbering for Use in a Genetic Algorithm Zhou and Gen =-=[9]-=- used Prufer numbers [4] to encode their potential solutions. This has two advantages: every Prufer number represents a valid spanning tree so that no repairing is necessary, and the degree of each ve... |

13 |
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Citation Context ...] report a network flow technique that improves upon these results, working again in linear time. They do not consider the more difficult non-Euclidean case, however. Recent research by Boldon et al. =-=[8]-=- describes a `dual simplex' approach, based on Prim's algorithm (the best in terms of speed and memory usage for finding the MST [1]), which produces good results on random, non-Euclidean graphs where... |

12 |
Edge exchanges in the degree-constrained minimum spanning tree problem
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Citation Context ...erge to a globally optimal solution. In their methods good upper bounds are generated by a modification to Prim's algorithm [4], which we refer to as d-Prim's in this paper. Savelsbergh and Volgenant =-=[5]-=- also employed a branch and bound technique with improved heuristics leading to considerably faster run times than those achieved by Narula and Ho. Random, non-Euclidean graphs were found to be far le... |

9 |
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Citation Context ...efer to the dual simplex method of Boldon et al. [8], using BF2 as the blacklisting function, as simply `BF2'. 3.4 Prufer Numbering for Use in a Genetic Algorithm Zhou and Gen [9] used Prufer numbers =-=[4]-=- to encode their potential solutions. This has two advantages: every Prufer number represents a valid spanning tree so that no repairing is necessary, and the degree of each vertex is represented expl... |

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3 |
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Citation Context ...ices left in U . Efficient implementation of Prim's algorithm, making good use of priority queues, leads to a running time of O(m log n) where m is the number of edges and n is the number of vertices =-=[14]-=-. There are three methods by which Prim's algorithm can be employed for finding solutions to the d-MST. The first method is to make an alteration to Prim's algorithm so that it does not add edges that... |