C. Walshaw. A Multilevel Approach to the Travelling Salesman Problem. To appear in Oper. Res., (originally published as Univ. Greenwich Tech. Rep. 00/IM/63), 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....to be more successful. The code is currently available from Simonetti s TSP webpage, http: www.contrib.andrew.cmu.edu neils tsp index.html. The second heuristic we cover also uses Chained LK to generate starting tours, but in a more sophisticated framework. A Multi Level Approach (Walshaw [71]) The idea is to recursively apply Chained LK (or any other local search heuristic) to a smaller Experimental Analysis of Heuristics for the STSP 65 coalesced instance in order to generate a starting tour for the full instance. A coalesced instance is created by matching nearby cities and ....

....on the previous instance as the starting tour for the current instance. We end up running the base heuristic Theta(log N) times, but most of the runs are on small instances, and overall running time is no more than 2 to 3 times that for running the base heuristic once on the full instance. See [71] for more details, including the geometry based method for matching cities during the coalescing phase. Walshaw submitted results for two instantiations of the Multi Level approach. The first (MLLK) used LK ABCC as its base heuristic and the second (MLCLK N) used CLK ABCC N. The base heuristics ....

C. Walshaw. A multilevel approach to the travelling salesman problem. Operations Research, to appear. Preliminary version available as Mathematics Research Report 00/1M/63, Computing and Mathematical Sciences Department, University of Greenwich.

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C. Walshaw. A Multilevel Approach to the Travelling Salesman Problem. To appear in Oper. Res., (originally published as Univ. Greenwich Tech. Rep. 00/IM/63), 2000.

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C. Walshaw. A Multilevel Approach to the Travelling Salesman Problem. Tech. Rep. 00/IM/63, Comp. Math. Sci., Univ. Greenwich, London SE10 9LS, UK, August 2000.

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C. Walshaw. A Multilevel Approach to the Travelling Salesman Problem. Accepted for Oper. Res., (originally published as Univ. Greenwich Tech. Rep. 00/IM/63), 2000.

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C. Walshaw. A Multilevel Approach to the Travelling Salesman Problem. To appear in Oper. Res., (originally published as Univ. Greenwich Tech. Rep. 00/IM/63), 2000.

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C. Walshaw. A Multilevel Approach to the Travelling Salesman Problem. Accepted for Oper. Res., (originally published as Univ. Greenwich Tech. Rep. 00/IM/63), 2000.

....work and list a number of possible methods for enhancing the techniques developed here. 2 Multilevel Optimisation Our interest in the multilevel paradigm arises from our work in the field of graph partitioning, e.g. 38] and subsequently graph drawing, 36] and the travelling salesman problem, [35]. Typically a P way graph partitioning algorithm aims to divide a graph into P disjoint subdomains of equal size and minimise the number of cut edges, an NP hard problem. In recent years it has been recognised that an effective way of both accelerating graph partitioning algorithms and, more ....

.... Kumar, 20] The multilevel partitioning strategy is widely used and forms the basis of several public domain partitioning packages including CHACO [14] JOSTLE [38] and METIS [20] More recently the multilevel paradigm has been applied with significant effect to the travelling salesman problem, [35]. The resulting algorithm progressively coarsens the problem, initialises a tour and then employs the Chained Lin Kernighan (CLK) algorithm to refine the solution on each of the coarsened problems in reverse order. In experiments on a well established test suite of 79 problem instances multilevel ....

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C. Walshaw. A Multilevel Approach to the Travelling Salesman Problem. Accepted for Oper. Res., (originally published as Univ. Greenwich Tech. Rep. 00/IM/63), 2000.

.... scheme, 59] With the realisation that this global quality might be put to good use for other discrete problems, and a possible explanation of the underlying process (at least for combinatorial optimisation problems) multilevel algorithms were then derived for the travelling salesman problem, [58], and for graph colouring, 60] In this paper, we aim to draw together some of this work, address the issue of multilevel refinement for combinatorial problems and, with the aid of examples in graph partitioning, graph colouring and the travelling salesman problem, make a case for its use as a ....

.... be Section 2 followed by Section 5 and then Sections 3 4, as the explanation and generalisation of the multilevel paradigm (Section 5) was originally derived from the multilevel partitioning examples (Section 2) and subsequently applied to derive multilevel approaches for the TSP (Section 3) in [58] and GCP (Section 4) in [60] However since the generalisation is also informed and clarified by examples from the TSP GCP, we have chosen to present the examples first and follow them with a generic overview. 1.2 Notation and definitions For all three example problems we use graph based ....

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C. Walshaw. A Multilevel Approach to the Travelling Salesman Problem. Accepted for Oper. Res., (originally published as Univ. Greenwich Tech. Rep. 00/IM/63), 2000.

....be specified as a complete graph with weighted edges, i.e. there is an edge between every pair of cities and the weight of the edge specifies the distance between them. 3. 1 A multilevel algorithm for the travelling salesman problem Recently the multilevel paradigm has been applied to the TSP, [16, 32, 33]. Although the scheme is perhaps less intuitive than multilevel partitioning, clearly the LK algorithm or one of its variants should make a good refinement method. However, with no graph as such, how can the problem be coarsened In fact from [32] it seems that the crucial point in devising a ....

....paradigm has been applied to the TSP, 16, 32, 33] Although the scheme is perhaps less intuitive than multilevel partitioning, clearly the LK algorithm or one of its variants should make a good refinement method. However, with no graph as such, how can the problem be coarsened In fact from [32] it seems that the crucial point in devising a coarsening algorithm is the requirement that the solution to each coarsened problem must contain a solution of the original problem (even if it is a poor solution) One way of achieving this is for the coarsening to successively fix edges into the ....

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C. Walshaw. A Multilevel Approach to the Travelling Salesman Problem. To appear in Oper. Res., (originally published as Univ. Greenwich Tech. Rep. 00/IM/63), 2000.

....results although it suffers from runtimes which are quadratic, in fact O(N 2:2 ) for problems of size N . In a further recent development, a general solution strategy known as multilevel refinement has been applied with considerable success 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 ....

....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 to be ....

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C. Walshaw. A Multilevel Approach to the Travelling Salesman Problem. To appear in Oper. Res., (originally published as Univ. Greenwich Tech. Rep. 00/IM/63), 2000.

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