The Challenge of Generating Spatially Balanced Scientific Experiment Designs

Abstract

The development of the theory and construction of combinatorial designs originated with the work of Euler on Latin squares. A Latin square on Ò symbols is an Ò¢Ò matrix (Ò is the order of the Latin square), in which each symbol occurs precisely once in each row and in each column. Several interesting research questions posed by Euler with respect to Latin squares, namely regarding orthogonality properties, were only solved in 1959 [3]. Many other questions concerning Latin squares constructions still remain open today.From the perspective of the Constraint Programing (CP), Artificial Intelligence (AI), and Operations Research (OR) communities, combinatorial design problems are interesting since they possess rich structural properties that are also observed in real-world applications such as scheduling, timetabling, and error correcting codes. Thus, the area of combinatorial designs has been a good source of challenge problems for these research communities. In fact, the study of combinatorial design problem instances has pushed the development of new search methods both in terms of systematic and stochastic procedures. For example, the question of the existence and non-existence of certain quasigroups (Latin squares) with intricate mathematical properties gives rise to some of the most challenging search problems in the context of automated theorem proving [16]. So-called general purpose model generation programs, used to prove theorems in finite domains, or to produce counterexamples to false conjectures, have been used to solve numerous previously open problems about the existence of Latin squares with specific mathematical properties. Considerable progress has also been made in the understanding of symmetry breaking procedures using benchmark problems based on combinatorial designs [5,6,9,13]. More recently, the study of search procedures on benchmarks based on Latin squares has led to the discovery of the non-standard probability distributions that characterize complete (randomized) backtrack search methods, so-called heavy-tailed distributions [8].

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