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Linear-size formulations for connected planar graph partitioning and political districting

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Abstract

Motivated by applications in political districting, we consider the task of partitioning the n vertices of a planar graph into k connected components. We propose an extended formulation for this task that has two desirable properties: (i) it uses just O(n) variables, constraints, and nonzeros, and (ii) it is perfect. To explore its ability to solve real-world problems, we apply it to a political districting problem in which contiguity and population balance are imposed as hard constraints and compactness is optimized. Computational experiments show that, despite the model’s small size and integrality for connected partitioning, the population balance constraints are more troublesome to effectively impose. Nevertheless, we share our findings in hopes that others may find better ways to impose them.

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Data availability

The datasets generated during and/or analysed during the current study are available at: https://github.com/JackDaihanZhang/Linear-size-formulations-for-connected-planar-graph-partitioning-and-political-districting.

References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice-Hall, Inc., Hoboken (1993)

    Google Scholar 

  2. Aprile, M., Fiorini, S., Huynh, T., Joret, G., Wood, D.R.: Smaller extended formulations for spanning tree polytopes in minor-closed classes and beyond. Electron. J. Comb. 28, P4-47 (2021)

    MathSciNet  Google Scholar 

  3. Carvajal, R., Constantino, M., Goycoolea, M., Vielma, J.P., Weintraub, A.: Imposing connectivity constraints in forest planning models. Oper. Res. 61(4), 824–836 (2013)

    Article  MathSciNet  Google Scholar 

  4. Cygan, M., Fomin, F.V., Kowalik, Ł, Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)

    Book  Google Scholar 

  5. DeFord, D., Najt, L., Stucky, E.: PlanarEmbeddings (2019). https://github.com/vrdi/NetworksWeek5/tree/master/PlanarEmbeddings

  6. Edmonds, J.: Optimum branchings. J. Res. Natl. Bureau Stand. B 71(4), 233–240 (1967)

    Article  MathSciNet  Google Scholar 

  7. Edmonds, J.: Matroids and the greedy algorithm. Math. Program. 1(1), 127–136 (1971)

    Article  MathSciNet  Google Scholar 

  8. Fiorini, S., Huynh, T., Joret, G., Pashkovich, K.: Smaller extended formulations for the spanning tree polytope of bounded-genus graphs. Discrete Comput. Geom. 57(3), 757–761 (2017)

    Article  MathSciNet  Google Scholar 

  9. Fischetti, M., Leitner, M., Ljubić, I., Luipersbeck, M., Monaci, M., Resch, M., Salvagnin, D., Sinnl, M.: Thinning out Steiner trees: a node-based model for uniform edge costs. Math. Program. Comput. 9(2), 203–229 (2017)

    Article  MathSciNet  Google Scholar 

  10. Garfinkel, R.S., Nemhauser, G.L.: Optimal political districting by implicit enumeration techniques. Manag. Sci. 16(8), B-495 (1970)

    Article  Google Scholar 

  11. Gibbons, A.: Algorithmic Graph Theory. Cambridge University Press, Cambridge (1985)

    Google Scholar 

  12. Haunert, J.H.: Aggregation in map generalization by combinatorial optimization. Ph.D. Thesis, Fachrichtung Geodäsie und Geoinformatik der Leibniz-Univ. (2008)

  13. Haunert, J.H., Wolff, A.: Generalization of land cover maps by mixed integer programming. In: Proceedings of the 14th Annual ACM International Symposium on Advances in Geographic Information Systems, pp. 75–82 (2006)

  14. Hebert, J.G., Vandenberg, M.E., Smith, P.: The Realist’s Guide to Redistricting: Avoiding the Legal Pitfalls. American Bar Association, Chicago (2010)

    Google Scholar 

  15. Hess, S., Weaver, J., Siegfeldt, H., Whelan, J., Zitlau, P.: Nonpartisan political redistricting by computer. Oper. Res. 13(6), 998–1006 (1965)

    Article  Google Scholar 

  16. Kim, M., Xiao, N.: Contiguity-based optimization models for political redistricting problems. Int. J. Appl. Geosp. Res. (IJAGR) 8(4), 1–18 (2017)

    Article  Google Scholar 

  17. Kim, M.J.: Multiobjective spanning tree based optimization model to political redistricting. Spat. Inf. Res. 26(3), 317–325 (2018)

    Article  Google Scholar 

  18. Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, Algorithms and Combinatorics, vol. 21, 6th edn. Springer, Berlin (2018)

    Book  Google Scholar 

  19. Martin, R.K.: Using separation algorithms to generate mixed integer model reformulations. Oper. Res. Lett. 10(3), 119–128 (1991)

    Article  MathSciNet  Google Scholar 

  20. NCSL: 2010 redistricting deviation table. https://www.ncsl.org/research/redistricting/2010-ncsl-redistricting-deviation-table.aspx (2020). Accessed 21 July 2022

  21. Oehrlein, J., Haunert, J.H.: A cutting-plane method for contiguity-constrained spatial aggregation. J. Sp. Inf. Sci. 15, 89–120 (2017)

    Google Scholar 

  22. Pashkovich, K.: Extended formulations for combinatorial polytopes. Ph.D. Thesis, Otto-von-Guericke-Universität Magdeburg (2012)

  23. Ricca, F., Scozzari, A., Simeone, B.: Political districting: from classical models to recent approaches. Ann. Oper. Res. 204(1), 271–299 (2013)

    Article  MathSciNet  Google Scholar 

  24. Ricca, F., Simeone, B.: Local search algorithms for political districting. Eur. J. Oper. Res. 189(3), 1409–1426 (2008)

    Article  Google Scholar 

  25. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)

    Google Scholar 

  26. Shirabe, T.: A model of contiguity for spatial unit allocation. Geogr. Anal. 37(1), 2–16 (2005)

    Article  Google Scholar 

  27. Shirabe, T.: Districting modeling with exact contiguity constraints. Environ. Plann. B. Plann. Des. 36(6), 1053–1066 (2009)

    Article  Google Scholar 

  28. Swamy, R., King, D.M., Jacobson, S.H.: Multiobjective optimization for politically fair districting: a scalable multilevel approach. Oper. Res. 71(2), 536–562 (2023)

    Article  MathSciNet  Google Scholar 

  29. Validi, H., Buchanan, A.: A note on “a linear-size zero-one programming model for the minimum spanning tree problem in planar graphs’’. Networks 73(1), 135–142 (2019)

    Article  MathSciNet  Google Scholar 

  30. Validi, H., Buchanan, A.: Political districting to minimize cut edges. Math. Program. Comput. 14(4), 623–672 (2022)

    Article  MathSciNet  Google Scholar 

  31. Validi, H., Buchanan, A., Lykhovyd, E.: Imposing contiguity constraints in political districting models. Oper. Res. 70(2), 867–892 (2022)

    Article  MathSciNet  Google Scholar 

  32. Wang, Y., Buchanan, A., Butenko, S.: On imposing connectivity constraints in integer programs. Math. Program. 166(1), 241–271 (2017)

    Article  MathSciNet  Google Scholar 

  33. Williams, J.C.: A linear-size zero-one programming model for the minimum spanning tree problem in planar graphs. Networks 39(1), 53–60 (2002)

    Article  MathSciNet  Google Scholar 

  34. Wong, R.T.: A dual ascent approach for Steiner tree problems on a directed graph. Math. Program. 28(3), 271–287 (1984)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This material is partially based upon work supported by the National Science Foundation under Grant No. 1942065 and by Rice University’s Building Research on Inequality and Diversity to Grow Equity (BRIDGE) seed Grant.

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Authors and Affiliations

  1. Operations Research Center, Massachusetts Institute of Technology, Cambridge, USA

    Jack Zhang

  2. Industrial, Manufacturing and Systems Engineering, Texas Tech University, Lubbock, USA

    Hamidreza Validi

  3. Industrial Engineering and Management, Oklahoma State University, Stillwater, USA

    Austin Buchanan

  4. Computational Applied Mathematics and Operations Research, Rice University, Houston, USA

    Illya V. Hicks

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  1. Jack Zhang
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  2. Hamidreza Validi
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  3. Austin Buchanan
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  4. Illya V. Hicks
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Correspondence to Austin Buchanan.

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Zhang, J., Validi, H., Buchanan, A. et al. Linear-size formulations for connected planar graph partitioning and political districting. Optim Lett 18, 19–31 (2024). https://doi.org/10.1007/s11590-023-02070-0

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