Proportional symbol maps are a cartographic tool to assist in the visualization and analysis of quantitative data associated with specific locations, such as earthquake magnitudes, oil well production, and temperature at weather stations. As the name suggests, symbol sizes are proportional to the magnitude of the physical quantities that they represent. We present two novel integer linear programming (ILP) models to solve this computational geometry problem: how to draw opaque disks on a map so as to maximize the total visible border of all disks. We focus on drawings obtained by layering symbols on top of each other, also known as stacking drawings. We introduce decomposition techniques, as well as several families of facet-defining inequalities which are used to strengthen the ILP models that are supplied to a commercial solver. We demonstrate the effectiveness of our approach through a series of computational experiments using hundreds of instances generated from real demographic and geophysical data sets. To the best of our knowledge, we are the first to use ILP to tackle this problem, and the first to provide provably optimal symbol maps for those data sets.

## Citation

INFORMS Journal on Computing, 2013. Published online before print DOI:10.1287/ijoc.2013.0557.