Combinatorial settlement planning

TL;DR

This paper studies a combinatorial model for settlement planning on grids, focusing on maximizing houses with sunlight exposure constraints, providing explicit calculations, bounds, and an integer programming approach.

Contribution

It introduces a new combinatorial model for settlement planning, deriving explicit formulas, bounds, and an integer programming formulation for maximal configurations.

Findings

Explicit calculation of minimum number of houses for given grid sizes

Bounds on maximum number of houses in maximal configurations

Integer programming formulation for small grid instances

Abstract

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 $m\times n$ 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 $n$.