C. Walshaw. Multilevel Refinement for Combinatorial Optimisation Problems. Tech. Rep. 01/IM/73, Comp. Math. Sci., Univ. Greenwich, London SE10 9LS, UK, June 2001. 8  Home/Search   Document Details and Download   Summary   Related Articles   Check

....be specifically designed to a problem, such that the coarsening process keeps the essence of the problem unchanged. In terms of its use for dealing with graph theoretic optimization problems, the multi scale approach is widely used for graph partitioning, see, e.g. 12] More recently, Walshaw [19] has used this approach for the TSP and vertex coloring problems. We have used the multi scale approach for the related problem of drawing graphs aesthetically [10] 14] 4.1 Segment Graphs One of the most prominent properties of multi scale algorithms is that they keep the inherent structure of ....

C. Walshaw, "Multilevel Refinement for Combinatorial Optimisation Problems", Technical Report 01/IM/73, Comp. Math. Sci., Univ. Greenwich, London, UK, 2001. 25

....for the particular problem at hand, so that the coarsening process keeps the essence of the problem unchanged. In terms of its use for dealing with graph theoretic optimization problems, the multi scale approach is widely used for graph partitioning, see, e.g. 14] More recently, Walshaw [21] has used this approach for the TSP and vertex coloring problems. We have used the multi scale approach for the related problem of drawing graphs aesthetically [12, 16] 9 4.1 Segment graphs One of the most prominent requirements of a good multi scale algorithm is that it keep the inherent ....

....that forces reasonable restrictions. On the other hand, coarsening based on graph structure is most often very local in nature, and may impose globally bad restrictions, like identifying vertices that should be very distant in the optimal solution. Solution based coarsening is discussed in [21]. The median iteration process we have proposed seems to have value in its own right. It is actually an extremely fast method for decreasing the cost of an arrangement, using a continuous relaxation of the original problem. It was applied successfully to graphs with millions of edges. ....

C. Walshaw, "Multilevel Refinement for Combinatorial Optimisation Problems", Technical Report 01/IM/73, Comp. Math. Sci., Univ. Greenwich, London, UK, 2001. A Proof of Lemma 2 We prove the following lemma, which appears in Subsection 4.2

....tailored for the particular problem at hand, so that the coarsening process keeps the essence of the problem unchanged. In terms of its use for dealing with graph theoretic optimization problems, the multiscale approach is widely used for graph partitioning, see, e.g. 7] More recently, Walshaw [13] has used this approach for the TSP and vertex coloring problems. We have used it for the related problem of drawing graphs aesthetically [5, 9] 4.1 Segment graphs One of the most prominent requirements of a good multi scale algorithm is that it keep the inherent structure of the problem ....

....solution for peforming a coarsening that forces reasonable restrictions. On the other hand, coarsening based on graph struc ture is most often very local in nature, and may impose globally bad restrictions, like identifying vertices that should be very distant in the optimal solution. See also [13]. The median iteration process we have proposed seems to have value in its own right. It is actually an extremely fast method for decreasing the cost of an arrangement, using a continuous relaxation of the original problem. In [10] it was applied successfully to graphs with millions of edges. ....

C. Walshaw, "Multilevel Refinement for Combinatorial Optimisation Problems", Technical Report 01/IM/73, Comp. Math. Sci., Univ. Greenwich, London, UK, 2001.

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C. Walshaw. Multilevel Refinement for Combinatorial Optimisation Problems. Tech. Rep. 01/IM/73, Comp. Math. Sci., Univ. Greenwich, London SE10 9LS, UK, June 2001. 8

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C. Walshaw. Multilevel Refinement for Combinatorial Optimisation Problems. Tech. Rep. 01/IM/73, Comp. Math. Sci., Univ. Greenwich, London SE10 9LS, UK, June 2001.

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C. Walshaw. Multilevel Refinement for Combinatorial Optimisation Problems. (To appear in Annals Oper. Res.; originally published as Univ. Greenwich Tech. Rep. 01/IM/73), 2001.

....to lower cost solutions (and often becoming trapped in local minima of the cost function) or upwards to higher cost. In this way the local search scheme induces a landscape on the solution space, so named because, conceptually at least, it consists of peaks, valleys and sometimes plateaus. In [34] the use of multilevel techniques in combinatorial optimisation is discussed with the aid of three example problems, graph partitioning, graph colouring and the travelling salesman problem. In that paper it is shown that, under certain conditions, multilevel coarsening is equivalent to recursively ....

....above, the coarsening constructs a series of approximations to the original problem; it is hoped that each problem P l retains the important features of its parent P l 1 but the (usually) randomised and irregular nature of the coarsening precludes any rigorous analysis of this process. In [34] this process is referred to as sampling the solution space; here we refer to it as filtering, partly because we wish to use the term sampling to describe another process in this paper, and more importantly, it is a more suggestive word for what actually occurs. On the other hand, viewing the ....

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C. Walshaw. Multilevel Refinement for Combinatorial Optimisation Problems. Tech. Rep. 01/IM/73, Comp. Math. Sci., Univ. Greenwich, London SE10 9LS, UK, June 2001.

.... to the TSP and in particular the CLK algorithm, 7, 13] The multilevel approach involves recursive coarsening to create a hierarchy of approximations to the original problem; an initial solution is found for the coarsest problem and then iteratively refined at each level, coarsest to finest, [14]. When applied to the TSP it was able to significantly enhance the performance of the CLK algorithm and in particular seemed to work much better for the more clustered problem instances with which TSP algorithms traditionally have great difficulties. In this paper we aim to bridge the gap between ....

....child is no help in solving the parent problem. Moreover, if the coarsening is constructed so as to sample the solution space, the resulting family of problems are simply restrictions of the original space rather than near approximations to it and this very much facilitates the solution process, [14]. In [13] a sampling based coarsening strategy for the TSP is derived which works by successively fixing edges into the tour. For example, given a TSP instance P of size N , if we fix an edge between cities c a and c b then we create a smaller problem P of size N 1 (because there are N 1 edges ....

[Article contains additional citation context not shown here]

C. Walshaw. Multilevel Refinement for Combinatorial Optimisation Problems. Tech. Rep. 01/IM/73, Comp. Math. Sci., Univ. Greenwich, London SE10 9LS, UK, June 2001. 8

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