Multilevel Landscapes in Combinatorial Optimisation

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.