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

Rafael G. Cano, Cid C. De Souza, Pedro J. De Rezende, Tallys Yunes

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)87-110
Number of pages24
JournalDiscrete Optimization
StatePublished - Nov 1 2015


  • Cartography
  • Computational geometry
  • Integer linear programming
  • Symbol maps
  • Visualization

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Applied Mathematics


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