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28
A MultiScale Algorithm for the Linear Arrangement Problem
 Proc. 28th Inter. Workshop on GraphTheoretic Concepts in Computer Science (WG’02), LNCS 2573
, 2002
"... Finding a linear ordering of the vertices of a graph is a common problem arising in diverse applications. In this paper we present a lineartime algorithm for this problem, based on the multiscale paradigm. Experimental results are similar to those of the best known approaches, while the running ti ..."
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Cited by 26 (4 self)
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Finding a linear ordering of the vertices of a graph is a common problem arising in diverse applications. In this paper we present a lineartime algorithm for this problem, based on the multiscale 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 multiscale construction, which may be used for a broader range of ordering problems.
JOSTLE: parallel multilevel graphpartitioning software – an overview
"... In this chapter we look at JOSTLE, the multilevel graphpartitioning 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 a ..."
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Cited by 14 (0 self)
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In this chapter we look at JOSTLE, the multilevel graphpartitioning 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 meshbased parallel scientific computing applications and so we also outline some enhancements such as multiphase meshpartitioning, heterogeneous mapping and partitioning to optimise subdomain shape.
Comparison of coarsening schemes for multilevel graph partitioning
 in: Learning and Intelligent Optimization: Third International Conference, LION 3. Selected Papers
, 2009
"... partitioning ..."
RELAXATIONBASED COARSENING AND MULTISCALE GRAPH ORGANIZATION
"... 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 ..."
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Cited by 5 (3 self)
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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.
A Multilevel Memetic Approach for Improving Graph Kpartitions
, 2011
"... 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 ef ..."
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Cited by 5 (3 self)
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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 perturbationbased 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.
CostBased Partitioning for Distributed and Parallel Simulation of Decomposable MultiScale Constructive Models. Simulation: Transactions of the Society for Modeling and Simulation International 82(12): 809–26
, 2006
"... Abstract. We present a concise, generic, and configurable new partitioning approach for decomposable, modular, and multiscale constructive models. A Generic Model Partitioning (GMP) algorithm decomposes a given multiscale model to a set of partition blocks based on a cost modeling and analysis met ..."
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Cited by 4 (0 self)
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Abstract. We present a concise, generic, and configurable new partitioning approach for decomposable, modular, and multiscale constructive models. A Generic Model Partitioning (GMP) algorithm decomposes a given multiscale model to a set of partition blocks based on a cost modeling and analysis method in polynomial time. It minimizes model decompositions and constructs monotonically improved partitioning outcomes during the partitioning process. The cost modeling and analysis method enables translating subjective, domainspecific, and heterogeneous resource information to objective, domainindependent, and homogeneous cost information. By translating models to a homogeneous cost space and describing partitioning logics over the space, the proposed algorithm utilizes domainspecific knowledge to produce the best partitioning results without any modification of its programming logics. As a consequence of its clean separation between domain specific partitioning requirements and goals, and generic partitioning logic, the proposed algorithm can be applied to a variety of partitioning problems in largescale systems biology research utilizing distributed and parallel simulation. It is expected that the algorithm improves overall performance and efficiency of in silico experimentation of complex multiscale biological system models.
An Effective Multilevel Tabu Search Approach for Balanced Graph Partitioning
, 2010
"... Graph partitioning is one of the fundamental NPcomplete 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 kpartitioning problem consists in partitioning the vertex set V into k disjoint subsets of about ..."
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Cited by 3 (1 self)
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Graph partitioning is one of the fundamental NPcomplete 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 kpartitioning 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 wellknown partitioning packages METIS and CHACO, but also improves more than two thirds of the best balanced partitions ever reported in the literature.
A Multilevel LinKernighanHelsgaun Algorithm for the Travelling Salesman Problem
 SE10 9LS
, 2001
"... 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 ..."
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Cited by 2 (1 self)
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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 LinKernighan (CLK) scheme was used for the refinement. However, a new and highly effective LinKernighan 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.
Lean algebraic multigrid (LAMG): Fast graph Laplacian linear solver
 ArXiV eprints
"... Abstract. Laplacian matrices of graphs arise in largescale computational applications such as semisupervised machine learning; spectral clustering of images, genetic data and web pages; transportation network flows; electrical resistor circuits; and elliptic partial differential equations discreti ..."
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Cited by 2 (0 self)
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Abstract. Laplacian matrices of graphs arise in largescale computational applications such as semisupervised machine learning; spectral clustering of images, genetic data and web pages; transportation network flows; electrical resistor circuits; and elliptic partial differential equations discretized on unstructured grids with finite elements. A Lean Algebraic Multigrid (LAMG) solver of the symmetric linear system Ax = b is presented, where A is a graph Laplacian. LAMG’s run time and storage are empirically demonstrated to scale linearly with the number of edges. LAMG consists of a setup phase during which a sequence of increasinglycoarser Laplacian systems is constructed, and an iterative solve phase using multigrid cycles. General graphs pose algorithmic challenges not encountered in traditional multigrid applications. LAMG combines a lean piecewiseconstant interpolation, judicious node aggregation based on a new node proximity measure (the affinity), and an energy correction of coarselevel systems. This results in fast convergence and substantial setup and memory savings. A serial LAMG implementation scaled linearly for a diverse set of 3774 realworld graphs with up to 47 million edges, with no parameter tuning. LAMG was more robust than the UMFPACK direct solver and Combinatorial Multigrid (CMG), although CMG was faster than LAMG on average. Our methodology is extensible to eigenproblems and other graph
Multilevel Landscapes in Combinatorial Optimisation
, 2002
"... 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 iter ..."
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Cited by 1 (1 self)
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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.