Spatially Balancing Polygon Grouping While Constraining Population Size

The Problem of Imbalanced Groupings

When dividing geographic areas into groups, a common objective is to create spatially balanced groupings while constraining the total population size in each group. However, imbalanced groupings often arise in practice due to complex spatial distributions of population and other factors.

Imbalanced spatial groupings can lead to unequal representation, access to resources, and allocation of public goods across groups. For example, election districts with widely varying populations may dilute voting power. Service districts like schools and hospitals may be overburdened if some groups contain far higher populations than others.

Therefore, it is important to develop robust methods that can spatially balance the division of polygons into groupings while constraining the population size as needed for particular applications.

Methods for Spatially Balancing Polygon Groups

Several computational methods exist for creating more spatially compact and balanced polygon groupings compared to manual or automated baselines:

  • Multi-objective evolutionary algorithms that optimize spatial compactness and population balance as objectives
  • Voronoi diagram approaches that partition space according to seed point locations
  • Graph partitioning techniques adapted to operate on polygons and geography
  • Hierarchical clustering methods that prioritize grouping adjacent polygons
  • Iterative grouping methods that swap polygons between groups to improve balance

More advanced methods also allow constraining the population size in polygon groups through various mechanisms, discussed next.

Constraining Group Sizes by Total Population

To ensure feasibility for downstream applications, polygon grouping methods must enable defining constraints on allowable group sizes based on total population:

  • Hard constraints strictly enforce minimum and/or maximum population thresholds per group
  • Soft constraints defined as optimization objectives or penalties to discourage threshold violations
  • Constraints checking and correction procedures, ensuring feasible groupings even given complex spatial distributions
  • Adaptive relaxation of constraints when infeasible spatially balanced groupings would otherwise result

By properly constraining group sizes by total population, balanced spatial groupings can be tailored to application requirements. But implementing the constraints introduces additional complexity.

Example Code for Spatially Balancing Polygon Grouping

Here is Python code for a spatial grouping method optimizing compactness and population balance under constraints:

import geopandas, pandas, sklearn

# Input geoDataFrame with population estimates per polygon 
gdf = geopandas.read_file(...)

# Define min/max threshold bounds per group
min_pop = 100000  
max_pop = 200000

# Generate initial random polygon grouping  
grouper = sklearn.Cluster(n_groups=10) 
groups = grouper.fit_predict(gdf)
gdf['group'] = groups

# Optimization iterations
for i in range(100):

  # Check/enforce population constraints 
  for group_id in
    pop = gdf[].population.sum()
    if pop < min_pop or pop > max_pop:
      repair_group(gdf, group_id)  

  #Swap polygons between groups to improve compactness 
  #and population balance  
  swap_pairs = find_boundary_swaps(gdf) 
  for a,b in swap_pairs:
    swap_group(gdf, a, b)

# Export final grouping  

This demonstrates a general template for iterative optimization and constraint handling. Many design choices and tuning considerations apply when implementing such methods in practice.

Evaluating the Quality of Spatially Balanced Groupings

To compare and select a polygon grouping method for an application, quantitative evaluation metrics should be computed on test datasets with known characteristics:

  • Compactness – Group spatial dispersion, perimeter-area ratios
  • Population balance – Statistical variance/deviation of group population totals
  • Constraint satisfaction – Percent of groups meeting given population thresholds
  • Bias metrics – Comparison to idealized district mappings given spatial distributions
  • Fairness metrics – Group-level similarity across socioeconomic attributes

Since enhancing one quality dimension can degrade others, evaluating tradeoffs is recommended. For example, strict population size constraints may reduce compactness and increase spatial clustering of disadvantaged communities.

Common Pitfalls and Challenges

Several common pitfalls should be avoided when designing and applying polygon grouping methods:

  • Assuming simple greedy and random heuristics will work at scale
  • Not testing algorithms on complex real-world geography and spatial population distributions
  • Insufficient tuning of parameters and constraints for a given region and application
  • Lack of quantitative quality measurement during method development
  • Overlooking demographic nuances, spatial autocorrelation, and side effects

Robustly balancing polygon groupings under population constraints remains an active area of research due to the complexity of underlying optimization problems.

Achieving Optimal Balance and Constraint Satisfaction

No universal guidelines guarantee optimally balancing polygon groups subject to constraints for all data and applications. However, the following practices help guide quality solutions:

  • Choose appropriate computational methods suited to problem nature
  • Leverage real-world geography/population data during design
  • Iteratively relax constraints as needed to achieve feasible balance
  • Visualize tradeoffs between quality dimensions
  • Quantitatively evaluate balance vs. compactness vs. constraints
  • Assess demographic qualities and spatial patterns in detail
  • Account for spatial autocorrelation and dependence

With diligent effort and testing, high-quality spatially balanced polygon groupings under population constraints can be constructed to serve downstream needs.


This overview covered the rationale, methods, tradeoffs, and practices for dividing geographic polygons into spatially compact groupings with constrained total population sizes per group. Challenges arise due to complex spatial distributions and interdependent quality dimensions, but incremental optimization and quantitative evaluation can produce robust solutions tailored to applications.

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