Combinatorial settlement planning

In this article, we consider a combinatorial settlement model on a rectangular grid where at least one side (east, south, or west) of each house must be exposed to sunlight without obstructions. We are interested in maximal configurations, where no additional houses can be added. For a fixed m x n grid, we explicitly calculate the lowest number of houses, and give close to optimal bounds on the highest number of houses that a maximal configuration can have. Additionally, we provide an integer programming formulation of the problem and solve it explicitly for small values of m and n.