Arc-based integer programming formulations for three variants of proportional symbol maps

Proportional symbol maps are a cartographic tool that employs scaled symbols to represent data associated with specific locations. The symbols we consider are opaque disks, which may be partially covered by other overlapping disks. We address the problem of creating a suitable drawing of the disks that maximizes one of two quality metrics: the total and the minimum visible length of disk boundaries. We study three variants of this problem, two of which are known to be NP-hard and another whose complexity is open. We propose novel integer programming formulations for each problem variant and test them on real-world instances with a branch-and-cut algorithm. When compared with state-of-the-art models from the literature, our models significantly reduce computation times for most instances. A new formulation is described to create optimal proportional symbol maps.We address three problem variants, two of which are known to be NP-hard.Efficient separation routines and lifting procedures are described.The new formulation is up to 82 times faster than the others in the literature.Most known benchmark instances can now be solved in less than one minute.