## A New Evolutionary Approach to the Degree Constrained Minimum Spanning Tree Problem (2000)

Venue: | IEEE Transactions on Evolutionary Computation |

Citations: | 11 - 0 self |

### BibTeX

@ARTICLE{Knowles00anew,

author = {Joshua Knowles and David Corne and Martin Oates},

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

journal = {IEEE Transactions on Evolutionary Computation},

year = {2000},

volume = {4},

pages = {125--134}

}

### OpenURL

### Abstract

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...

### Citations

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Citation Context ...ry algorithm (EA) approach which uses it. Section 6 records the results achieved when comparing Boldon et al.'s method, dPrim 's, and the RPM (multistart hillclimbing, simulated annealing [8], and EA =-=[9, 10, 11]-=- versions) on thirteen random graphs with a range of degree constraints. This also compares our evolutionary approach with a recently published alternative evolutionary approach [13] on a test problem... |

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Citation Context ...ry algorithm (EA) approach which uses it. Section 6 records the results achieved when comparing Boldon et al.'s method, dPrim 's, and the RPM (multistart hillclimbing, simulated annealing [8], and EA =-=[9, 10, 11]-=- versions) on thirteen random graphs with a range of degree constraints. This also compares our evolutionary approach with a recently published alternative evolutionary approach [13] on a test problem... |

612 |
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Citation Context ...ry algorithm (EA) approach which uses it. Section 6 records the results achieved when comparing Boldon et al.'s method, dPrim 's, and the RPM (multistart hillclimbing, simulated annealing [8], and EA =-=[9, 10, 11]-=- versions) on thirteen random graphs with a range of degree constraints. This also compares our evolutionary approach with a recently published alternative evolutionary approach [13] on a test problem... |

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Citation Context ...n the initialisation phase. The recombination operator used is uniform crossover. These two operators were applied in an EA using a steady-state version of Collins and Jeffersons' Local Mating method =-=[7]-=-, with a population of 225 (a 15 \Theta 15 grid) and a random-walk length of 3. For comparison, the initialisation and mutation operations were also used in a hill climber and a simulated annealing al... |

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Citation Context ...dean graph, the unconstrained MST never contains vertices of degree greater than five. In fact, for Euclidean graphs it has been shown that the d-MST is NP-hard when d != 4 but polynomial when d ?= 5 =-=[5]-=-. In a random graph, by contrast, the maximum degree of a vertex contained in the unconstrained MST can be arbitrarily large (up to N-1): Ravi et al. [5] state that given any rational number R ?= 1, f... |

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Citation Context ...onary 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... |

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Citation Context ...ard test problem. 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. However, in real networks the vertices (or nodes) are usually subject to a degree constraint. For example, exchange... |

18 |
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Citation Context ...omised Primal Method) runs in reasonable time and significantly outperforms both d-Prim's and Boldon et al.'s method in terms of solution quality. It also outperforms a recent alternative EA approach =-=[13]-=-. All methods are compared on a variety of randomly generated networks of from 50 nodes to 3000 nodes. The random graphs used for testing have been created to be particularly difficult and misleading ... |

13 |
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Citation Context ...e exchange failure. Unlike the MST, the general d-MST is NP-hard; notice that when d = 2, it is equivalent to the traveling salesperson problem without the last edge. Recent research by Boldon et al. =-=[2]-=- 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 seems to perform very well on a range of benchmark d-MST pr... |

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Citation Context ...epresentation based on Prufer numbering, proposed by Zhou and Gen [13]. To this end, we present results on a nine node Euclidean Graph which was first given as an example by Savelsbergh and Volgenant =-=[12]-=-, who solved it heuristically, and was used by Zhou and Gen to show that their EA could also solve it. The edge weights of the graph are given in Table 4. The optimal solution to the graph is 2256 whe... |

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Citation Context ...g 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 ... |

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Citation Context ...r evolutionary algorithm (EA) approach which uses it. Section 6 records the results achieved when comparing Boldon et al.'s method, dPrim 's, and the RPM (multistart hillclimbing, simulated annealing =-=[8]-=-, and EA [9, 10, 11] versions) on thirteen random graphs with a range of degree constraints. This also compares our evolutionary approach with a recently published alternative evolutionary approach [1... |