## Biased mutation operators for subgraph-selection problems (2004)

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Venue: | IN IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION |

Citations: | 2 - 0 self |

### BibTeX

@ARTICLE{Raidl04biasedmutation,

author = {Günther R. Raidl and Gabriele Koller and Bryant A. Julstrom},

title = {Biased mutation operators for subgraph-selection problems},

journal = {IN IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION},

year = {2004}

}

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### Abstract

Many graph problems seek subgraphs of minimum weight that satisfy a set of constraints. Examples include the minimum spanning tree problem (MSTP), the degree-constrained minimum spanning tree problem (d-MSTP), and the traveling salesman problem (TSP). Low-weight 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 edge-exchange mutation. In a large-scale empirical investigation, we study the distributions of edges in optimum solutions of the MSTP, the d-MSTP, 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

### Citations

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Citation Context ...regard it. Further, replacing a by |S|/(|S| + 1) according to (5), we obtain: EX ∗ (e ∗ ) ≈ a |S| (1 − √ = a) 2 = � 1 (|S| + 1) 1 − = � �2 |S| |S|+1 = �� � �2 |S| + |S| + 1 . (19) By Chernoff’s bound =-=[14]-=-, the probability of deviations from this expected value is exponentially small with respect to n; that is, the number of edge-selections required lies in the interval [EX ∗ (e∗ )/2, 2 · EX ∗ (e∗ )] w... |

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Citation Context ...e approximately optimal mutation scheme, its expected running time on an average instance of the MSTP is asymptotically not worse than the time of Kruskal’s well-known minimum spanning tree algorithm =-=[12]-=-. Section V describes a variety of other strategies with which mutation may select edges for inclusion in new solutions, and Sect. VI describes experiments with the (1+1)-EA for the MSTP that confirm ... |

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Citation Context ...cting the solutions in their initial populations and to their recombination and mutation operators, which construct new solutions from existing ones. Several researchers have examined such mechanisms =-=[6]-=-, [7], [8], [9]. Among them, Julstrom and Raidl studied weightbiased crossover operators in EAs for the TSP and the dMSTP on complete graphs [10]; favoring low-weight edges improved the performance of... |

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Citation Context ...so that each MST is unique. The structure of the (1+1)-EA is standard. Its initial solution is an unbiased random spanning tree, generated via a random walk in the target graph as described by Broder =-=[16]-=-. The expected time of this step is O(n log n) for almost all graphs, including in particular complete graphs. Each iteration of the EA applies mutation to its current solution to create one offspring... |

66 | Randomized local search, evolutionary algorithms, and the minimum spanning tree problem
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Citation Context ...e expected times that a (1+1)- EA requires to find an unconstrained MST on a complete graph G = (V, E), using five different edge-selection and replacement strategies in mutation. Neumann and Wegener =-=[15]-=- investigated simple randomized local search (RLS) and an unbiased (1+1)-EA for the MSTP on general (including incomplete) graphs. Their algorithms encoded candidate subgraphs as bit-strings that indi... |

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Citation Context ...ces with n = 100 or fewer vertices were considered. The corresponding 3-MSTP, 5-MSTP, and TSP instances were solved to optimality by branch-and-cut algorithms implemented using the ABACUS environment =-=[13]-=- and CPLEX 8.1 as a linear programming solver. In each instance, sorting the edges into ascending order of their weights assigns each a rank r, 1 ≤ r ≤ m; ties are broken arbitrarily. Figure 1 plots t... |

37 |
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Citation Context ...hat its time t NA mut is Θ(1), the other methods run in linear time in the worst case, when T is a path. The diameter of a random spanning tree on G, however, is Θ( √ n) with overwhelming probability =-=[17]-=-, [18], and thus, on average, these mutation variants run in times tUU mut, tUG mut, tBU mut, and tBG mut that are all O( √ n). To identify the expected times the (1+1)-EA, with each mutation operator... |

24 |
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Citation Context ...ning tree problem; MSTP). • S is a spanning tree in which the number of edges incident on each vertex does not exceed a bound d > 1 (the degree-constrained minimum spanning tree problem; d-MSTP) [1], =-=[2]-=-. • S is a spanning tree with at least L leaves (the leafconstrained minimum spanning tree problem) [3], [4]. • S augments a given subgraph so that the resulting network is biconnected (the biconnecti... |

15 | Edge-sets: An effective evolutionary coding of spanning trees
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Citation Context ...solutions in their initial populations and to their recombination and mutation operators, which construct new solutions from existing ones. Several researchers have examined such mechanisms [6], [7], =-=[8]-=-, [9]. Among them, Julstrom and Raidl studied weightbiased crossover operators in EAs for the TSP and the dMSTP on complete graphs [10]; favoring low-weight edges improved the performance of these alg... |

13 |
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Citation Context ...s time t NA mut is Θ(1), the other methods run in linear time in the worst case, when T is a path. The diameter of a random spanning tree on G, however, is Θ( √ n) with overwhelming probability [17], =-=[18]-=-, and thus, on average, these mutation variants run in times tUU mut, tUG mut, tBU mut, and tBG mut that are all O( √ n). To identify the expected times the (1+1)-EA, with each mutation operator, requ... |

11 | A new evolutionary approach to the degree constrained minimum spanning tree problem
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Citation Context ... spanning tree problem; MSTP). • S is a spanning tree in which the number of edges incident on each vertex does not exceed a bound d > 1 (the degree-constrained minimum spanning tree problem; d-MSTP) =-=[1]-=-, [2]. • S is a spanning tree with at least L leaves (the leafconstrained minimum spanning tree problem) [3], [4]. • S augments a given subgraph so that the resulting network is biconnected (the bicon... |

10 |
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Citation Context ... the solutions in their initial populations and to their recombination and mutation operators, which construct new solutions from existing ones. Several researchers have examined such mechanisms [6], =-=[7]-=-, [8], [9]. Among them, Julstrom and Raidl studied weightbiased crossover operators in EAs for the TSP and the dMSTP on complete graphs [10]; favoring low-weight edges improved the performance of thes... |

6 | A memetic algorithm for minimum-cost vertex-biconnectivity augmentation of graphs
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Citation Context ...th at least L leaves (the leafconstrained minimum spanning tree problem) [3], [4]. • S augments a given subgraph so that the resulting network is biconnected (the biconnectivity augmentation problem) =-=[5]-=-. • S is a Steiner tree that connects a specified subset of G’s vertices. Some of these problems, such as the unconstrained MSTP and the identification of a shortest path between two vertices, can be ... |

6 | Weight-biased edge-crossover in evolutionary algorithms for two graph problems
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- 2001
(Show Context)
Citation Context ...ones. Several researchers have examined such mechanisms [6], [7], [8], [9]. Among them, Julstrom and Raidl studied weightbiased crossover operators in EAs for the TSP and the dMSTP on complete graphs =-=[10]-=-; favoring low-weight edges improved the performance of these algorithms. The present authors investigated weight-biased mutation in these EAs and derived probabilities for selecting edges that minimi... |

4 |
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(Show Context)
Citation Context ...oes not exceed a bound d > 1 (the degree-constrained minimum spanning tree problem; d-MSTP) [1], [2]. • S is a spanning tree with at least L leaves (the leafconstrained minimum spanning tree problem) =-=[3]-=-, [4]. • S augments a given subgraph so that the resulting network is biconnected (the biconnectivity augmentation problem) [5]. • S is a Steiner tree that connects a specified subset of G’s vertices.... |

2 | Codings and operators in two genetic algorithms for the leaf-constrained minimum spanning tree problem
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- 2004
(Show Context)
Citation Context ...ot exceed a bound d > 1 (the degree-constrained minimum spanning tree problem; d-MSTP) [1], [2]. • S is a spanning tree with at least L leaves (the leafconstrained minimum spanning tree problem) [3], =-=[4]-=-. • S augments a given subgraph so that the resulting network is biconnected (the biconnectivity augmentation problem) [5]. • S is a Steiner tree that connects a specified subset of G’s vertices. Some... |

1 |
On the bias and performance of the edgeset encoding
- Rothlauf, Tzschoppe
- 2004
(Show Context)
Citation Context ...ions in their initial populations and to their recombination and mutation operators, which construct new solutions from existing ones. Several researchers have examined such mechanisms [6], [7], [8], =-=[9]-=-. Among them, Julstrom and Raidl studied weightbiased crossover operators in EAs for the TSP and the dMSTP on complete graphs [10]; favoring low-weight edges improved the performance of these algorith... |

1 |
On weight-biased mutation for graph problems,” in Parallel Problem Solving from Nature
- Raidl, Kodydek, et al.
- 2002
(Show Context)
Citation Context ...orithms. The present authors investigated weight-biased mutation in these EAs and derived probabilities for selecting edges that minimize the expected time to include edges of optimum tours and trees =-=[11]-=-. This article extends in several ways our work on biased mutation in EAs for subset-selection problems on complete graphs. The next section investigates empirically the distributions of edges in opti... |