On the Computational Tractability of a Geographic Clustering Problem Arising in Redistricting

Authors Vincent Cohen-Addad, Philip N. Klein, Dániel Marx, Archer Wheeler, Christopher Wolfram

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Vincent Cohen-Addad
  • CNRS and Sorbonne Université, Paris, France
Philip N. Klein
  • Brown University, Providence, RI, USA
Dániel Marx
  • CISPA Helmholtz Center for Information Security, Saarland Informatics Campus, Germany
Archer Wheeler
  • Brown University, Providence, RI, USA
Christopher Wolfram
  • Brown University, Providence, RI, USA

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Vincent Cohen-Addad, Philip N. Klein, Dániel Marx, Archer Wheeler, and Christopher Wolfram. On the Computational Tractability of a Geographic Clustering Problem Arising in Redistricting. In 2nd Symposium on Foundations of Responsible Computing (FORC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 192, pp. 3:1-3:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Redistricting is the problem of dividing up a state into a given number k of regions (called districts) where the voters in each district are to elect a representative. The three primary criteria are: that each district be connected, that the populations of the districts be equal (or nearly equal), and that the districts are "compact". There are multiple competing definitions of compactness, usually minimizing some quantity. One measure that has been recently been used is number of cut edges. In this formulation of redistricting, one is given atomic regions out of which each district must be built (e.g., in the U.S., census blocks). The populations of the atomic regions are given. Consider the graph with one vertex per atomic region and an edge between atomic regions with a shared boundary of positive length. Define the weight of a vertex to be the population of the corresponding region. A districting plan is a partition of vertices into k pieces so that the parts have nearly equal weights and each part is connected. The districts are considered compact to the extent that the plan minimizes the number of edges crossing between different parts. There are two natural computational problems: find the most compact districting plan, and sample districting plans (possibly under a compactness constraint) uniformly at random. Both problems are NP-hard so we consider restricting the input graph to have branchwidth at most w. (A planar graph’s branchwidth is bounded, for example, by its diameter.) If both k and w are bounded by constants, the problems are solvable in polynomial time. In this paper, we give lower and upper bounds that characterize the complexity of these problems in terms of parameters k and w. For simplicity of notation, assume that each vertex has unit weight. We would ideally like algorithms whose running times are of the form O(f(k,w) n^c) for some constant c independent of k and w (in which case the problems are said to be fixed-parameter tractable with respect to those parameters). We show that, under standard complexity-theoretic assumptions, no such algorithms exist. However, the problems are fixed-parameter tractable with respect to each of these parameters individually: there exist algorithms with running times of the form O(f(k) n^{O(w)}) and O(f(w) n^{k+1}). The first result was previously known. The new one, however, is more relevant to the application to redistricting, at least for coarse instances. Indeed, we have implemented a version of the algorithm and have used to successfully find optimally compact solutions to all redistricting instances for France (except Paris, which operates under different rules) under various population-balance constraints. For these instances, the values for w are modest and the values for k are very small.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • redistricting
  • algorithms
  • planar graphs
  • lower bounds


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  1. Sachet Bangia, Christy Vaughn Graves, Gregory Herschlag, Han Sung Kang, Justin Luo, Jonathan C. Mattingly, and Robert Ravier. Redistricting: Drawing the line, 2017. URL: http://arxiv.org/abs/1704.03360.
  2. Hans L. Bodlaender, Daniel Lokshtanov, and Eelko Penninkx. Planar capacitated dominating set is W[1]-hard. In Jianer Chen and Fedor V. Fomin, editors, Proceedings of the 4th International Workshop on Parameterized and Exact Computation, volume 5917 of Lecture Notes in Computer Science, pages 50-60. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-11269-0_4.
  3. Daniel Carter, Gregory Herschlag, Zach Hunter, and Jonathan Mattingly. A merge-split proposal for reversible Monte Carlo Markov Chain sampling of redistricting plans, 2019. URL: http://arxiv.org/abs/1911.01503.
  4. J. Chen. Expert report of Jowei Chen, Ph.D., Raleigh Wake Citizen’s Association et al. vs. the Wake County Board of Elections, 2017. URL: https://www.pubintlaw.org/wp-content/uploads/2017/06/Expert-Report-Jowei-Chen.pdf.
  5. Vincent Cohen-Addad, Philip N. Klein, and Neal E. Young. Balanced centroidal power diagrams for redistricting. In Proceedings of the 26th ACM International Conference on Advances in Geographic Information Systems, pages 389-396, 2018. URL: https://doi.org/10.1145/3274895.3274979.
  6. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Daniel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 1st edition, 2015. Google Scholar
  7. Daryl DeFord and Moon Duchin. Redistricting reform in Virginia: Districting criteria in context. Virginia Policy Review, 12(2):120-146, 2019. Google Scholar
  8. Daryl DeFord, Moon Duchin, and Justin Solomon. Recombination: A family of Markov chains for redistricting, 2019. URL: http://arxiv.org/abs/1911.05725.
  9. Michael Dom, Daniel Lokshtanov, Saket Saurabh, and Yngve Villanger. Capacitated domination and covering: A parameterized perspective. In Proceedings of the 3rd International WorkshopParameterized and Exact Computation, volume 5018 of Lecture Notes in Computer Science, pages 78-90. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-79723-4_9.
  10. Frederic Dorn, Eelko Penninkx, Hans L. Bodlaender, and Fedor V. Fomin. Efficient exact algorithms on planar graphs: Exploiting sphere cut decompositions. Algorithmica, 58(3):790-810, 2010. URL: https://doi.org/10.1007/s00453-009-9296-1.
  11. Moon Duchin. Geography meets geometry in redistricting. Conference at Center for Geographic Analysis at Harvard University, May 2019. URL: https://cga-download.hmdc.harvard.edu/publish_web/CGA_Conferences/2019_Redistricting/slides/Moon_Duchin.pdf.
  12. Moon Duchin and Bridget Eileen Tenner. Discrete geometry for electoral geography, 2018. URL: http://arxiv.org/abs/1808.05860.
  13. Martin E. Dyer and Alan M. Frieze. On the complexity of partitioning graphs into connected subgraphs. Discret. Appl. Math., 10(2):139-153, 1985. URL: https://doi.org/10.1016/0166-218X(85)90008-3.
  14. David Eppstein, Michael T. Goodrich, Doruk Korkmaz, and Nil Mamano. Defining equitable geographic districts in road networks via stable matching. In Proceedings of the 25th ACM International Conference on Advances in Geographic Information Systems, pages 52:1-52:4, 2017. URL: https://doi.org/10.1145/3139958.3140015.
  15. Andreas Emil Feldmann and Dániel Marx. The parameterized hardness of the k-center problem in transportation networks. Algorithmica, 82(7):1989-2005, 2020. URL: https://doi.org/10.1007/s00453-020-00683-w.
  16. Michael R. Fellows, Fedor V. Fomin, Daniel Lokshtanov, Frances A. Rosamond, Saket Saurabh, Stefan Szeider, and Carsten Thomassen. On the complexity of some colorful problems parameterized by treewidth. Inf. Comput., 209(2):143-153, 2011. Google Scholar
  17. Kyle Fox, Philip N. Klein, and Shay Mozes. A polynomial-time bicriteria approximation scheme for planar bisection. In Proceedings of the 47th Annual ACM on Symposium on Theory of Computing, pages 841-850, 2015. URL: https://doi.org/10.1145/2746539.2746564.
  18. R. S. Garfinkel and G. L. Nemhauser. Optimal political districting by implicit enumeration techniques. Management Science, 16(8):B-495, 1970. Google Scholar
  19. Naveen Garg, Huzur Saran, and Vijay V. Vazirani. Finding separator cuts in planar graphs within twice the optimal. SIAM J. Comput., 29(1):159-179, 1999. URL: https://doi.org/10.1137/S0097539794271692.
  20. Sushmita Gupta, Saket Saurabh, and Meirav Zehavi. On treewidth and stable marriage. CoRR, abs/1707.05404, 2017. URL: http://arxiv.org/abs/1707.05404.
  21. Gregory Z. Gutin, Mark Jones, and Magnus Wahlström. The mixed Chinese postman problem parameterized by pathwidth and treedepth. SIAM J. Discret. Math., 30(4):2177-2205, 2016. URL: https://doi.org/10.1137/15M1034337.
  22. Robert E Helbig, Patrick K Orr, and Robert R Roediger. Political redistricting by computer. Communications of the ACM, 15(8):735-741, 1972. Google Scholar
  23. Gregory Herschlag, Han Sung Kang, Justin Luo, Christy Vaughn Graves, Sachet Bangia, Robert Ravier, and Jonathan C. Mattingly. Quantifying gerrymandering in North Carolina, 2018. URL: http://arxiv.org/abs/1801.03783.
  24. Gregory Herschlag, Robert Ravier, and Jonathan C. Mattingly. Evaluating partisan gerrymandering in Wisconsin, 2017. URL: http://arxiv.org/abs/1709.01596.
  25. S. W. Hess, J. B. Weaver, H. J. Siegfeldt, J. N. Whelan, and P. A. Zitlau. Nonpartisan political redistricting by computer. Operations Research, 13(6):998-1006, 1965. Google Scholar
  26. Takehiro Ito, Kazuya Goto, Xiao Zhou, and Takao Nishizeki. Partitioning a multi-weighted graph to connected subgraphs of almost uniform size. IEICE Trans. Inf. Syst., 90-D(2):449-456, 2007. URL: https://doi.org/10.1093/ietisy/e90-d.2.449.
  27. Takehiro Ito, Xiao Zhou, and Takao Nishizeki. Partitioning a graph of bounded tree-width to connected subgraphs of almost uniform size. J. Discrete Algorithms, 4(1):142-154, 2006. URL: https://doi.org/10.1016/j.jda.2005.01.005.
  28. Klaus Jansen, Stefan Kratsch, Dániel Marx, and Ildikó Schlotter. Bin packing with fixed number of bins revisited. J. Comput. Syst. Sci., 79(1):39-49, 2013. URL: https://doi.org/10.1016/j.jcss.2012.04.004.
  29. Jun Kawahara, Takashi Horiyama, Keisuke Hotta, and Shin-ichi Minato. Generating all patterns of graph partitions within a disparity bound. In International Workshop on Algorithms and Computation, pages 119-131. Springer, 2017. Google Scholar
  30. Philip N. Klein and Shay Mozes. Optimization Algorithms for Planar Graphs. http://planarity.org/. accessed June 2018.
  31. Dániel Marx, Ario Salmasi, and Anastasios Sidiropoulos. Constant-factorapproximations for asymmetric TSP on nearly-embeddable graphs. In Proceedings of the 19th Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, volume 60 of LIPIcs, pages 16:1-16:54, 2016. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.16.
  32. Anuj Mehrotra, Ellis L. Johnson, and George L. Nemhauser. An optimization based heuristic for political districting. Management Science, 44(8):1100-1114, 1998. Google Scholar
  33. Lorenzo Najt, Daryl R. DeFord, and Justin Solomon. Complexity and geometry of sampling connected graph partitions. CoRR, abs/1908.08881, 2019. URL: http://arxiv.org/abs/1908.08881.
  34. J. K. Park and C. A. Phillips. Finding minimum-quotient cuts in planar graphs. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing, pages 766-775, 1993. URL: https://doi.org/10.1145/167088.167284.
  35. W. Pegden. Pennsylvania’s congressional districting is an outlier: Expert report, League of Women Voters vs. Pennsylvania General Assembly, 2017. URL: https://www.brennancenter.org/sites/default/files/legal-work/LWV_v_PA_Expert_Report_WesleyPegden_11.17.17.pdf.
  36. Richard H Pildes, Tacy F Flint, and Sidley Austin. Brief of political geography scholars as amici curiae in support of appellees. URL: https://www.brennancenter.org/sites/default/files/legal-work/Gill_AmicusBrief_%20Political%20Geography%20Scholars_InSupportofAppellees.pdf.
  37. Satish Rao. Finding near optimal separators in planar graphs. In Proceedings of the 28th Annual IEEE Symposium on Foundations of Computer Science, pages 225-237, 1987. URL: https://doi.org/10.1109/SFCS.1987.26.
  38. Satish Rao. Faster algorithms for finding small edge cuts in planar graphs. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing, pages 229-240, 1992. URL: https://doi.org/10.1145/129712.129735.