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Optimizing Administrative Divisions: A Vertex $k$-Center Approach for Edge-Weighted Road Graphs

Peteris Daugulis

TL;DR

The paper tackles equitable and efficient administrative-territorial division by minimizing travel-time disparities on an edge-weighted road graph using a Voronoi partition around centers and a minimax vertex $k$-center formulation, demonstrated on Latvia. It introduces a constrained center-in-subgraph condition $c\in Z(G[V_S(c)])$, and develops a two-stage approximation (greedy farthest clustering plus center shifting with local search) to produce balanced territorial units (TUs) with borders visualized via alpha shapes or cross-edge methods. Applied to Latvia, the method shows substantial potential: reducing the number of TUs by up to $58\%$ while keeping the maximal TU radius comparable to or lower than current divisions, and highlighting substantial travel-time inequality in the present system. The work provides a transparent, data-driven framework for reform with clear policy implications and a path to incorporate additional weights and socio-economic factors in future iterations.

Abstract

Efficient and equitable access to municipal services hinges on well-designed administrative divisions. It requires ongoing adaptation to changing demographics, infrastructure, and economic factors. This article proposes a novel transparent data-driven method for territorial division based on the Voronoi partition of edge-weighted road graphs and the vertex $k$-center problem as a special case of the minimax facility location problem. By considering road network structure and strategic placement of administrative centers, this method seeks to minimize travel time disparities and ensure a more balanced distribution of administrative time burden for the population. We show implementations of this approach in the context of Latvia, a country with complex geographical features and diverse population distribution.

Optimizing Administrative Divisions: A Vertex $k$-Center Approach for Edge-Weighted Road Graphs

TL;DR

The paper tackles equitable and efficient administrative-territorial division by minimizing travel-time disparities on an edge-weighted road graph using a Voronoi partition around centers and a minimax vertex -center formulation, demonstrated on Latvia. It introduces a constrained center-in-subgraph condition , and develops a two-stage approximation (greedy farthest clustering plus center shifting with local search) to produce balanced territorial units (TUs) with borders visualized via alpha shapes or cross-edge methods. Applied to Latvia, the method shows substantial potential: reducing the number of TUs by up to while keeping the maximal TU radius comparable to or lower than current divisions, and highlighting substantial travel-time inequality in the present system. The work provides a transparent, data-driven framework for reform with clear policy implications and a path to incorporate additional weights and socio-economic factors in future iterations.

Abstract

Efficient and equitable access to municipal services hinges on well-designed administrative divisions. It requires ongoing adaptation to changing demographics, infrastructure, and economic factors. This article proposes a novel transparent data-driven method for territorial division based on the Voronoi partition of edge-weighted road graphs and the vertex -center problem as a special case of the minimax facility location problem. By considering road network structure and strategic placement of administrative centers, this method seeks to minimize travel time disparities and ensure a more balanced distribution of administrative time burden for the population. We show implementations of this approach in the context of Latvia, a country with complex geographical features and diverse population distribution.

Paper Structure

This paper contains 19 sections, 1 theorem, 2 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methods
  3. Data
  4. Computations
  5. From centers to TUs in road graphs - Voronoi cells in edge-weighted graphs
  6. Optimal centers - a minimax facility location problem
  7. A modified vertex $k$-center problem and its application to territorial division
  8. Approximation algorithms for the territorial division problem
  9. Borders of TUs
  10. Results and discussion
  11. Properties of the road graph of Latvia
  12. Examples of territorial division
  13. $k=1$
  14. $k>1$
  15. $k=5$
  16. ...and 4 more sections

Key Result

proposition thmcounterproposition

Let $G=(V,E,w)$ be an undirected edge-weighted graph with a positive weight function, $S\subseteq V$, $\textbf{A}=\{(V_S(c),c)\}_{c\in S}$ - the centered Voronoi partition for $S$. Then for any centered partition $\textbf{B}=\{(V'_{c},c)\}_{c\in S}$ with $c\in V'_a$ we have $r(\textbf{A})\le r(\text

Figures (5)

  • Figure 1: $\Gamma$ - a partial road graph of Latvia.
  • Figure 2: The case $k=1$. The black dot is in the geographic position of $Z(\Gamma)$. Red lines are state main roads.
  • Figure 3: Cases $2\le k\le 9$. The alpha shape view of TUs. The letters are at the geographic positions of the centers. Red lines are preliminary borders of TUs.
  • Figure 4: A case $k=5$.
  • Figure 5: A case $k=15$. The alpha shape view.

Theorems & Definitions (2)

  • proposition thmcounterproposition
  • proof