Generating optimal drawings of physically realizable symbol maps with integer programming
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Proportional symbol maps are a tool often used by cartographers and geoscience professionals to visualize geopositioned data associated with events and demographic statistics, such as earthquakes and population counts. Symbols are placed at specific locations on a map, and their areas are scaled to become proportional to the magnitudes of the data points they represent. We focus specifically on creating physically realizable drawings of symbols—opaque disks, in our case—by maximizing two quality metrics: the total and the minimum length of their visible borders. As these two maximization problems have been proven to be NP-hard, we provide integer programming formulations for their solution, along with decomposition techniques designed to decrease the size of input instances. Our computational experiments, which use real-life data sets, demonstrate the effectiveness of our approach and provide, for the first time, a number of optimal solutions to previously studied instances of this problem.
KeywordsVisualization Cartography Computational geometry Integer linear programming
Guilherme Kunigami is supported by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) grant 830510/1999-0. Pedro J. de Rezende is partially supported by CNPq grants 483177/2009-1, 473867/2010-9, FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) grant 07/52015-0, and a grant from FAEPEX/UNICAMP. Cid C. de Souza is partially supported by CNPq grants 301732/2007-8, 472504/2007-0, 473867/2010-9, and FAPESP grant 07/52015-0.
The authors would like to express their appreciation to the anonymous referees for their careful and thorough reviews and for their comments which contributed to improving the exposition.
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