Clustering solutions: a novel approach to solving NP-complete problems

Abstract

In this thesis, we introduce a novel approach to solving MAX-SAT problems. This algorithm clusters good solutions, and restarts the search from the closest feasible configuration to the centroid of each cluster. We call this method Clustered-Landscape Guided Hopping (CLGH). In addition, where clustering does not provide an advantage due to the non-clustered landscape configuration, we use Averaged-Landscape Guided Hopping (ALGH). CLGH is shown to be highly efficient for finding good solutions of large MAX-SAT problems. Systematic studies of the landscape are presented to show that the success of clustering is due to the learning of large-scale structure of the fitness landscape. Previous studies conducted by other researchers analysed the relationship between local and global minima and provided an insight into the configuration of the landscape. It was found that local minima formed clusters around global ones. We expanded these analyses to cover the relationship between clusters, and found that local minima form many correlated yet distant clusters. In addition, we show the existence of a relationship between the size of the problem and the distance between local minima. To rule out other possibilities of this success we test several other population based algorithms, and compare their performances to clustering. In addition, we compare with solo-search algorithms. We show that this method is superior to all algorithms tested. CLGH produces results that might be produced by a solo-local search algorithm within 95% less time. However, this is not a standalone technique, and can be incorporated within other algorithms to further enhance their performance. A further application of clustering is carried out on the Traveling Salesman Problem (TSP) in the discrete domain, and Artificial Neural Networks (ANN) using backpropagation for the purpose of data classification in the continuous domain. Since TSP does not show a clustered landscape configuration we find that ALGH is an effective method for improving search results. Preliminary results are shown indicating that extensions of the proposed algorithm can give similar improvements on these hard optimisation problems.

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