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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.

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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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