Mathematical Programming

, Volume 149, Issue 1–2, pp 425–457 | Cite as

Efficient geo-graph contiguity and hole algorithms for geographic zoning and dynamic plane graph partitioning

  • Douglas M. King
  • Sheldon H. Jacobson
  • Edward C. Sewell
Full Length Paper Series A


Graph partitioning is an intractable problem that arises in many practical applications. Heuristics such as local search generate good (though suboptimal) solutions in limited time. Such heuristics must be able to explore the solution space quickly and, when the solution space is constrained, differentiate feasible solutions from infeasible ones. Geographic zoning problems allocate some resource (e.g., political representation, school enrollment, police patrols) to contiguous zones modeled by partitions of an embedded planar graph. Each vertex corresponds to an area of the plane (e.g., census block, town, county), and local search moves one area from its current zone to a different zone in each iteration. Enforcing contiguity constraints may require significant computation when the graph is large. While existing algorithms require linear or polylogarithmic time (in the number of vertices) to assess contiguity in each local search iteration, the geo-graph paradigm shows how contiguity can be verified by examining only the set of vertices that border the transferred area (i.e., those areas whose boundaries share at least a single point with the boundary of the transferred area). This paper develops efficient algorithms that examine these vertices more quickly than traditional search-based methods, allowing practitioners to more fully consider their zoning options when creating zones with local search.


Graph partitioning Planar graphs Graph contiguity  Political districting 

Mathematics Subject Classification (2000)

05C85, Graph algorithms 05C40, Connectivity 05C10, Planar graphs 05A18, Partitions of sets 


  1. 1.
    Alpert, C.J., Kahng, A.B.: Recent directions in netlist partitioning: a survey. INTEGRATION. VLSI J. 19, 1–81 (1995)CrossRefzbMATHGoogle Scholar
  2. 2.
    Barth, D., Pellegrini, F., Raspaud, A., Roman, J.: On bandwidth, cutwidth, and quotient graphs. Theor. Inform. Appl. 29(6), 487–508 (1995)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Becker, R.I., Lari, I., Lucertini, M., Simeone, B.: Max-min partitioning of grid graphs into connected components. Networks 32(2), 115–125 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bender, M.A., Farach-Colton, M.: The level ancestor problem simplified. Theor. Comput. Sci. 321(1), 5–12 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bozkaya, B., Erkut, E., Laporte, G.: A tabu search heuristic and adaptive memory procedure for political districting. Eur. J. Oper. Res. 144(1), 12–26 (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Buchsbaum, A.L., Georgiadis, L., Kaplan, H., Rogers, A., Tarjan, R.E., Westbrook, J.R.: Linear-time algorithms for dominators and other path-evaluation problems. SIAM J. Comput. 38(4), 1533–1573 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Butler, D., Cain, B.E.: Congressional Redistricting: Comparative and Theoretical Perspectives. MacMillan Publishing Company, New York (1992)Google Scholar
  8. 8.
    Chlebíková, J.: Approximating the maximally balanced connected partition problem in graphs. Inf. Process. Lett. 60, 225–230 (1996)CrossRefGoogle Scholar
  9. 9.
    di Cortona, P.G., Manzi, C., Pennisi, A., Ricca, F., Simeone, B.: Evaluation and Optimization of Electoral Systems. Society for Industrial and Applied Mathematics, Philadelphia (1999)CrossRefzbMATHGoogle Scholar
  10. 10.
    D’Amico, S., Wang, S., Batta, R., Rump, C.: A simulated annealing approach to police district design. Comput. Oper. Res. 29, 667–684 (2002)CrossRefzbMATHGoogle Scholar
  11. 11.
    Demetrescu, C., Eppstein, D., Galil, Z., Italiano, G.F.: Dynamic graph algorithms. In: Atallah, M.J., Blanton, M. (eds.) Algorithms and Theory of Computation Handbook, 2nd edn. pp. 9.1–9.28. Chapman & Hall/CRC, London (2010)Google Scholar
  12. 12.
    Eppstein, D., Galil, Z., Italiano, G.F., Spencer, T.H.: Separator based sparsification for dynamic planar graph algorithms. In: Proceedings of the 25th Annual ACM Symposium on Theory of Computing, pp. 208–217. ACM, New York, NY (1993)Google Scholar
  13. 13.
    Eppstein, D., Italiano, G.F., Tamassia, R., Tarjan, R.E., Westbrook, J., Yung, M.: Maintenance of a minimum spanning forest in a dynamic plane graph. J. Algorithms 13(1), 33–54 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Fiduccia, C.M., Mattheyses, R.M.: A linear-time heuristic for improving network partitions. In: DAC ’82: Proceedings of the 19th Design Automation Conference, pp. 175–181. IEEE Press, Piscataway, NJ (1982)Google Scholar
  15. 15.
    Firmani, D., Italiano, G.F., Laura, L., Orlandi, A., Santaroni, F.: Computing strong articulation points and strong bridges in large scale graphs. In: Klasing, R. (ed.) Experimental Algorithms: Proceedings of the 11th International Symposium, SEA 2012, Bordeaux, France, June 7–9, 2012, pp. 195–207. Springer, Berlin (2012)Google Scholar
  16. 16.
    Frigioni, D., Italiano, G.F.: Dynamically switching vertices in planar graphs. Algorithmica 28(1), 76–103 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Henzinger, M.R., King, V.: Randomized fully dynamic graph algorithms with polylogarithmic time per operation. J. ACM 46(4), 502–516 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Henzinger, M.R., King, V.: Maintaining minimum spanning forests in dynamic graphs. SIAM J. Comput. 31(2), 364–374 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Holm, J., de Lichtenberg, K., Thorup, M.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM 48(4), 723–760 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Horowitz, E., Sahni, S.: Fundamentals of Computer Algorithms. Computer Science Press, Rockville (1978)zbMATHGoogle Scholar
  21. 21.
    Kernighan, B.W., Lin, S.: An efficient heuristic procedure for partitioning graphs. Bell Syst. Tech. J. 49(1), 291–307 (1970)CrossRefzbMATHGoogle Scholar
  22. 22.
    King, D.M., Jacobson, S.H., Sewell, E.C., Cho, W.K.T.: Geo-graphs: an efficient model for enforcing contiguity and hole constraints in planar graph partitioning. Oper. Res. 60(5), 1213–1228 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Lengauer, T., Tarjan, R.E.: A fast algorithm for finding dominators in a flowgraph. ACM Trans. Program. Lang. Syst. 1(1), 121–141 (1979)CrossRefzbMATHGoogle Scholar
  24. 24.
    Reif, J.H.: A topological approach to dynamic graph connectivity. Inf. Process. Lett. 25(1), 65–70 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Ricca, F.: A multicriteria districting heuristic for the aggregation of zones and its use in computing origin-destination matrices. INFOR 42(1), 61–77 (2004)MathSciNetGoogle Scholar
  26. 26.
    Ricca, F., Scozzari, A., Simeone, B.: Political districting: from classical models to recent approaches. 4OR 9(3), 223–254 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Ricca, F., Simeone, B.: Local search algorithms for political districting. Eur. J. Oper. Res. 189, 1409–1426 (2008)CrossRefzbMATHGoogle Scholar
  28. 28.
    Salgado, L.R.B., Wakabayashi, Y.: Approximation results on balanced connected partitions of graphs. Electron. Notes Discret. Math. 18, 207–212 (2004)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)CrossRefGoogle Scholar
  30. 30.
    Tarjan, R.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Tavares-Pereira, F., Figueira, J.R., Mousseau, V., Roy, B.: Multiple criteria districting problems: the public transportation network pricing system of the Paris region. Ann. Oper. Res. 154, 69–92 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Thorup, M.: Near-optimal fully-dynamic graph connectivity. In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pp. 343–350. ACM, New York, NY (2000)Google Scholar
  33. 33.
    United States Census Bureau: Apportionment Population and Number of Representatives, by State: Census 2000. (2000). Accessibility Verified on 27 May 2011
  34. 34.
    United States Census Bureau: Tallies of Census Blocks By State or State Equivalent. (2011). Accessibility Verified on 27 May 2011
  35. 35.
    Wang, J.P.: Stochastic relaxation on partitions with connected components and its application to image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 20(6), 619–636 (1998)CrossRefGoogle Scholar
  36. 36.
    West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice Hall, Upper Saddle River (2001)Google Scholar
  37. 37.
    Whitney, H.: Non-separable and planar graphs. Trans. Am. Math. Soc. 34(2), 339–362 (1932)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  • Douglas M. King
    • 1
  • Sheldon H. Jacobson
    • 2
  • Edward C. Sewell
    • 3
  1. 1.Department of Industrial and Enterprise Systems EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Department of Computer ScienceUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.Department of Mathematics and StatisticsSouthern Illinois University EdwardsvilleEdwardsvilleUSA

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