The BIN_COUNTS Constraint: Filtering and Applications

TL;DR

This paper introduces the BIN_COUNTS constraint for counting decision variables in specified bins, along with algorithms for filtering and applications including statistical testing and workload balancing.

Contribution

It presents the BIN_COUNTS constraint, a filtering algorithm for generalized arc consistency, and a new statistical constraint based on the $ ext{chi}^2$ test, with diverse practical applications.

Findings

Filtering algorithms improve constraint propagation.

The $ ext{chi}^2$ test constraint aids in statistical analysis.

Applications demonstrate effectiveness in workload and curriculum problems.

Abstract

We introduce the BIN_COUNTS constraint, which deals with the problem of counting the number of decision variables in a set which are assigned values that lie in given bins. We illustrate a decomposition and a filtering algorithm that achieves generalised arc consistency. We contrast the filtering power of these two approaches and we discuss a number of applications. We show that BIN_COUNTS can be employed to develop a decomposition for the $\chi^2$ test constraint, a new statistical constraint that we introduce in this work. We also show how this new constraint can be employed in the context of the Balanced Academic Curriculum Problem and of the Balanced Nursing Workload Problem. For both these problems we carry out numerical studies involving our reformulations. Finally, we present a further application of the $\chi^2$ test constraint in the context of confidence interval analysis.