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.

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.

A policy-definition language and prototype implementation library for policy-based autonomic systems

This paper presents work towards generic policy toolkit support for autonomic computing systems in which the policies themselves can be adapted dynamically and automatically. The work is motivated by three needs: the need for longer-term policy-based adaptation where the policy itself is dynamically adapted to continually maintain or improve its effectiveness despite changing environmental conditions; the need to enable non autonomics-expert practitioners to embed self-managing behaviours with low cost and risk; and the need for adaptive policy mechanisms that are easy to deploy into legacy code. A policy definition language is presented; designed to permit powerful expression of self-managing behaviours. The language is very flexible through the use of simple yet expressive syntax and semantics, and facilitates a very diverse policy behaviour space through both hierarchical and recursive uses of language elements. A prototype library implementation of the policy support mechanisms is described. The library reads and writes policies in well-formed XML script. The implementation extends the state of the art in policy-based autonomics through innovations which include support for multiple policy versions of a given policy type, multiple configuration templates, and meta-policies to dynamically select between policy instances and templates. Most significantly, the scheme supports hot-swapping between policy instances. To illustrate the feasibility and generalised applicability of these tools, two dissimilar example deployment scenarios are examined. The first is taken from an exploratory implementation of self-managing parallel processing, and is used to demonstrate the simple and efficient use of the tools. The second example demonstrates more-advanced functionality, in the context of an envisioned multi-policy stock trading scheme which is sensitive to environmental volatility

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.