Experimental Development of Automated Search Techniques for Discrete Combinatorial Optimisation

Abstract

A suite of techniques for finding the optimal solutions for a set of discrete combinatorial problems was developed. An experimental approach was used, with a suitable test-bed found in a class of word-puzzles. The crux of such research is that seeking optimal solutions to discrete combinatorial problems requires the use of deterministic algorithms. Attention was focused on the development of new techniques capable of exhausting the search space more efficiently. Although research was restricted to tractable problems, exhaustion of the search space was recognised to be practically infeasible for all but small problem instances. Thus the size and complexity of the problems examined was necessarily restricted. On these grounds the selection of an appropriate test-bed was fundamental to the research. Complex word problems were used because they encompass a wide range of discrete combinatorial problems, but have only a small literature. The specific puzzle examples employed as test-beds had all been used in public competitions with solutions submitted by thousands of humans, with the winning solutions and scores published. This allowed a simple and independent initial benchmark of success. The techniques developed could be judged to be at least partially successful in that they were able to at least equal and in some cases beat the highest recorded scores. The general problem of benchmarking is discussed. It was observed that small changes to the test bed puzzles or to the techniques would often impact dramatically on the results. In an attempt to isolate the reasons for this, a focused view of the search algorithms was adopted. Complex holistic algorithms were broken into smaller sub-algorithmic categories, such as: node selection, domain maintenance, forward tracking, backtracking, branch-and-bound, primary slot selection, variable ordering, value ordering, and constraint ordering. Within each of these categories a range of variations is presented. Techniques for removing inconsistencies prior to search were also experimented with. These consistency pre-processors were found to have a minimal and at times detrimental effect on search times when a good selection of search techniques was used. However, they were found to offer considerable benefits in instances where a poor selection of search techniques was chosen. As such these consistency pre-processors may be viewed as useful in terms of a risk management strategy for solving these problems. Whilst not the primary focus of this research experimentation with stochastic techniques within a deterministic framework was performed. The purpose of which was to gauge the impact of generating good solutions prior to an exhaustive search. A technique developed was observed to frequently improve the time taken to form an optimal solution, and improve the total time taken to exhaust the search space. While the major effort in the research was necessarily spent in developing and testing these algorithms and their implementations, specific attention was paid to the methodological problems inherent in experimental approaches to program development.