Finding a linear ordering of the vertices of a graph is a common problem arising in diverse applications. In this paper we present a linear-time algorithm for this problem, based on the multi-scale paradigm. Experimental results are similar to those of the best known approaches, while the running time is significantly better, enabling it to deal with much larger graphs. The paper contains a general multi-scale construction, which may be used for a broader range of ordering problems.

partitioning

In this chapter we look at JOSTLE, the multilevel graph-partitioning software package, and highlight some of the key research issues that it addresses. We first outline the core algorithms and place it in the context of the multilevel refinement paradigm. We then look at issues relating to its use as a tool for parallel processing and, in particular, partitioning in parallel. Since its first release in 1995, JOSTLE has been used for many mesh-based parallel scientific computing applications and so we also outline some enhancements such as multiphase mesh-partitioning, heterogeneous mapping and partitioning to optimise subdomain shape.

The multilevel paradigm has recently been applied to the travelling salesman problem with considerable success. The resulting algorithm progressively coarsens the problem, initialises a tour and then employs a local search algorithm to refine the solution on each of the coarsened problems in reverse order. In the original version the chained Lin-Kernighan (CLK) scheme was used for the refinement. However, a new and highly effective Lin-Kernighan variant (LKH) has recently been developed by Helsgaun. Here then we report on the modifications required to develop a multilevel LKH algorithm and the results achieved. Although the LKH algorithm, with its extremely high quality results, is more difficult to improve on than the CLK, nonetheless the multilevel framework was able to enhance the LKH performance. For example, in experiments on a well established test suite, the multilevel LKH scheme found 39 out of 59 optimal solutions as compared to the 33 found by LKH in a similar time period.

Graph partitioning is one of the most studied NPcomplete problems. Given a graph G = (V, E), the task is to partition the vertex set V into k disjoint subsets of about the same size, such that the number of edges with endpoints in different subsets is minimized. In this work, we present a highly effective multilevel memetic algorithm, which integrates a new multiparent crossover operator and a powerful perturbation-based tabu search algorithm. The proposed crossover operator tends to preserve the backbone with respect to a certain number of parent individuals, i.e. the grouping of vertices which is common to all parent individuals. Extensive experimental studies on numerous benchmark instances from the Graph Partitioning Archive show that the proposed approach, within a time limit ranging from several minutes to several hours, performs far better than any of the existing graph partitioning algorithm in terms of solution quality.

In this paper we generalize and improve the multiscale organization of graphs by introducing a new measure that quantifies the “closeness” between two nodes. The calculation of the measure is linear in the number of edges in the graph and involves just a small number of relaxation sweeps. A similar notion of distance is then calculated and used at each coarser level. We demonstrate the use of this measure in multiscale methods for several important combinatorial optimization problems and discuss the multiscale graph organization.

We consider the multilevel paradigm and its potential to aid the solution of combinatorial optimisation problems. The multilevel paradigm is a simple one, which involves recursive coarsening to create a hierarchy of approximations to the original problem. An initial solution is found and then iteratively refined at each level, coarsest to finest. Although the multilevel paradigm has been in use for many years and has been applied to many problem areas (most notably in the form of multigrid methods), it has only recently been suggested as a metaheuristic for combinatorial optimisation problems. It has been proposed that, for such problems, multilevel coarsening is equivalent to recursively filtering the solution space to create a hierarchy of increasingly coarser and smaller spaces. It is also suggested that perhaps this aids the local search algorithms used to refine the solution on each level by somehow `smoothing' the landscape of the solution spaces. In this paper, with some example problem instances drawn from graph partitioning and the travelling salesman problem, we take a detailed look at how the coarsening affects the hierarchy of solution landscapes. In particular we are interested in how the coarsening and hence filtering of the original space impacts on the maximum, minimum and average values of the cost function in the coarsened spaces. However, we also explore the manner in which the density of problem instances can moderate the effectiveness of a multilevel refinement algorithm.

Abstract. The Traveling Salesman Problem (TSP) is a well-known NPhard combinatorial optimization problem, for which a large variety of evolutionary algorithms are known. However, these heuristics fail to find solutions for large instances due to time and memory constraints. Here, we discuss a set of edge fixing heuristics to transform large TSP problems into smaller problems, which can be solved easily with existing algorithms. We argue, that after expanding a reduced tour back to the original instance, the result is nearly as good as applying the used solver to the original problem instance, but requiring significantly less time to be achieved. We claim that with these reductions, very large TSP instances can be tackled with current state-of-the-art evolutionary local search heuristics. 1

Graph partitioning is one of the fundamental NP-complete problems which is widely applied in many domains, such as VLSI design, image segmentation, data mining etc. Given a graph G =(V,E), the balanced k-partitioning problem consists in partitioning the vertex set V into k disjoint subsets of about the same size, such that the number of cutting edges is minimized. In this paper, we present a multilevel algorithm for balanced partition, which integrates a powerful refinement procedure based on tabu search with periodic perturbations. Experimental evaluations on a wide collection of benchmark graphs show that the proposed approach not only competes very favorably with the two well-known partitioning packages METIS and CHACO, but also improves more than two thirds of the best balanced partitions ever reported in the literature.

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