Multilevel mesh partitioning for heterogeneous communication networks

Multilevel algorithms are a successful class of optimisation techniques which address the mesh partitioning problem for distributing unstructured meshes onto parallel computers. They usually combine a graph contraction algorithm together with a local optimisation method which refines the partition at each graph level. To date these algorithms have been used almost exclusively to minimise the cut edge weight in the graph with the aim of minimising the parallel communication overhead, but recently there has been a perceived need to take into account the communications network of the parallel machine. For example the increasing use of SMP clusters (systems of multiprocessor compute nodes with very fast intra-node communications but relatively slow inter-node networks) suggest the use of hierarchical network models. Indeed this requirement is exacerbated in the early experiments with meta-computers (multiple supercomputers combined together, in extreme cases over inter-continental networks). In this paper therefore, we modify a multilevel algorithm in order to minimise a cost function based on a model of the communications network. Several network models and variants of the algorithm are tested and we establish that it is possible to successfully guide the optimisation to reflect the chosen architecture.

A multilevel algorithm for force-directed graph drawing

We describe a heuristic method for drawing graphs which uses a multilevel technique combined with a force-directed placement algorithm. The multilevel process groups vertices to form clusters, uses the clusters to define a new graph and is repeated until the graph size falls below some threshold. The coarsest graph is then given an initial layout and the layout is successively refined on all the graphs starting with the coarsest and ending with the original. In this way the multilevel algorithm both accelerates and gives a more global quality to the force- directed placement. The algorithm can compute both 2 & 3 dimensional layouts and we demonstrate it on a number of examples ranging from 500 to 225,000 vertices. It is also very fast and can compute a 2D layout of a sparse graph in around 30 seconds for a 10,000 vertex graph to around 10 minutes for the largest graph. This is an order of magnitude faster than recent implementations of force-directed placement algorithms.

A multilevel approach to the travelling salesman problem

We motivate, derive, and implement a multilevel approach to the travelling salesman problem.The resulting algorithm progressively coarsens the problem, initialises a tour, and then employs either the Lin-Kernighan (LK) or 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 80 problem instances we found multilevel configurations that either improved the tour quality by over 25% as compared to the standard CLK algorithm using the same amount of execution time, or that achieved approximately the same tour quality over seven times more rapidly. Moreover, the multilevel variants seem to optimise far better the more clustered instances with which the LK and CLK algorithms have the most difficulties.

A multilevel algorithm for force-directed graph-drawing

We describe a heuristic method for drawing graphs which uses a multilevel framework combined with a force-directed placement algorithm. The multilevel technique matches and coalesces pairs of adjacent vertices to define a new graph and is repeated recursively to create a hierarchy of increasingly coarse graphs, G0, G1, …, GL. The coarsest graph, GL, is then given an initial layout and the layout is refined and extended to all the graphs starting with the coarsest and ending with the original. At each successive change of level, l, the initial layout for Gl is taken from its coarser and smaller child graph, Gl+1, and refined using force-directed placement. In this way the multilevel framework both accelerates and appears to give a more global quality to the drawing. The algorithm can compute both 2 & 3 dimensional layouts and we demonstrate it on examples ranging in size from 10 to 225,000 vertices. It is also very fast and can compute a 2D layout of a sparse graph in around 12 seconds for a 10,000 vertex graph to around 5-7 minutes for the largest graphs. This is an order of magnitude faster than recent implementations of force-directed placement algorithms.

Multilevel refinement for combinatorial optimisation problems

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 (sometimes for the original problem, sometimes the coarsest) and then iteratively refined at each level. As a general solution strategy, 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 techniques). However, with the exception of the graph partitioning problem, multilevel techniques have not been widely applied to combinatorial optimisation problems. In this paper we address the issue of multilevel refinement for such problems and, with the aid of examples and results in graph partitioning, graph colouring and the travelling salesman problem, make a case for its use as a metaheuristic. The results provide compelling evidence that, although the multilevel framework cannot be considered as a panacea for combinatorial problems, it can provide an extremely useful addition to the combinatorial optimisation toolkit. We also give a possible explanation for the underlying process and extract some generic guidelines for its future use on other combinatorial problems.

A combined evolutionary search and multilevel optimisation approach to graph-partitioning

The graph-partitioning problem is to divide a graph into several pieces so that the number of vertices in each piece is the same within some defined tolerance and the number of cut edges is minimised. Important applications of the problem arise, for example, in parallel processing where data sets need to be distributed across the memory of a parallel machine. Very effective heuristic algorithms have been developed for this problem which run in real-time, but it is not known how good the partitions are since the problem is, in general, NP-complete. This paper reports an evolutionary search algorithm for finding benchmark partitions. A distinctive feature is the use of a multilevel heuristic algorithm to provide an effective crossover. The technique is tested on several example graphs and it is demonstrated that our method can achieve extremely high quality partitions significantly better than those found by the state-of-the-art graph-partitioning packages.

Multilevel refinement for the vehicle routing problem

Multilevel approaches to computational problems are pervasive across many areas of applied mathematics and scientific computing. The multilevel paradigm uses recursive coarsening to create a hierarchy of approximations to the original problem, then an initial solution is found for the coarsest problem and iteratively refined and improved at each level, coarsest to finest. The solution process is aided by the global perspective (or `global view') imparted to the optimisation by the coarsening. This paper looks at their application to the Vehicle Routing Problem.

JOSTLE: multilevel graph partitioning software: an overview

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

Multilevel refinement strategies for the capacity vehicle routing problem

We discuss the application of the multilevel (ML) refinement technique to the Vehicle Routing Problem (VRP), and compare it to its single-level (SL) counterpart. Multilevel refinement recursively coarsens to create a hierarchy of approximations to the problem and refines at each level. A SL algorithm, which uses a combination of standard VRP heuristics, is developed first to solve instances of the VRP. A ML version, which extends the global view of these heuristics, is then created, using variants of the construction and improvement heuristics at each level. Finally some multilevel enhancements are developed. Experimentation is used to find suitable parameter settings and the final version is tested on two well-known VRP benchmark suites. Results comparing both SL and ML algorithms are presented.

The application of multilevel refinement to the vehicle routing problem

We discuss the application of the multilevel (ML) refinement technique to the Vehicle Routing Problem (VRP), and compare it to its single-level (SL) counterpart. Multilevel refinement recursively coarsens to create a hierarchy of approximations to the problem and refines at each level. A SL heuristic, termed the combined node-exchange composite heuristic (CNCH), is developed first to solve instances of the VRP. A ML version (the ML-CNCH) is then created, using the construction and improvement heuristics of the CNCH at each level. Experimentation is used to find a suitable combination, which extends the global view of these heuristics. Results comparing both SL and ML are presented.

Multilevel refinement for combinatorial optimisation: boosting metaheuristic performance

The multilevel paradigm as applied to combinatorial optimisation problems is a simple one, which at its most basic involves recursive coarsening to create a hierarchy of approximations to the original problem. An initial solution is found, usually at the coarsest level, and then iteratively refined at each level, coarsest to finest, typically by using some kind of heuristic optimisation algorithm (either a problem-specific local search scheme or a metaheuristic). Solution extension (or projection) operators can transfer the solution from one level to another. As a general solution strategy, the multilevel paradigm has been in use for many years and has been applied to many problem areas (for example multigrid techniques can be viewed as a prime example of the paradigm). Overview papers such as [] attest to its efficacy. However, with the exception of the graph partitioning problem, multilevel techniques have not been widely applied to combinatorial problems and in this chapter we discuss recent developments. In this chapter we survey the use of multilevel combinatorial techniques and consider their ability to boost the performance of (meta)heuristic optimisation algorithms.

Multilevel approaches applied to the capacitated clustering problem

This paper presents two multilevel refinement algorithms for the capacitated clustering problem. Multilevel refinement is a collaborative technique capable of significantly aiding the solution process for optimisation problems. The central methodologies of the technique are filtering solutions from the search space and reducing the level of problem detail to be considered at each level of the solution process. The first multilevel algorithm uses a simple tabu search while the other executes a standard local search procedure. Both algorithms demonstrate that the multilevel technique is capable of aiding the solution process for this combinatorial optimisation problem.