Packing Compact Subgraphs with Applications to Districting

2026-04-10

AI Summary

This work addresses the problem of efficiently packing disjoint subgraphs in planar graphs, apex-minor-free graphs, and bounded-expansion graphs to maximize coverage while satisfying connectivity, constant-radius (e.g., stars or radius-$k$ districts), and composition constraints such as minimum population or political balance. Leveraging graph packing and approximation techniques, the study improves the approximation factor for balanced star-shaped districting from $O(\log n)$ to a constant and extends this result to broader graph classes. By allowing slight relaxation of balance requirements, the approach achieves a $(1+\varepsilon)$-approximation for coverage. The paper also establishes hardness-of-approximation lower bounds for related variants and presents constant-factor approximation algorithms that effectively balance constraint satisfaction and coverage performance across multiple graph families.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Packing disjoint subgraphs in a given graph is a fundamental problem with many applications. Motivated by political districting, we focus on connected subgraphs that are compact (e.g., having constant radius from a single center vertex) and that satisfy additional composition requirements, such as a minimum population/weight threshold or balanced weight types (e.g., political affiliations). We aim to maximize coverage by disjoint districts that meet these requirements. In this work, we present new results that substantially improve the previously known bounds on balanced star districts for planar and minor-free graphs (Dharangutte et al. 2025). In particular, we improve the approximation factor from $O(\log n)$ to $O(1)$ for packing balanced star districts using the exact same algorithm, but with a refined analysis. We also extend the results beyond planar graphs to minor-free graphs and an even broader family of graphs of bounded expansion. Additionally, we obtain an $O(1)$ approximation for packing radius-$k$ districts (with a constant $k$) in planar and apex-minor-free graphs. In order to get a $(1+\varepsilon)$ approximation on the max coverage, we show that this can be achieved if we allow a slight relaxation of the balancedness parameters (by a factor that can be made arbitrarily close to $1$), for bounded radius-$k$ districts on planar and apex-minor-free graphs. We show that all of these results can also be obtained if we enforce a minimum weight threshold for each district as the composition requirement, rather than balancedness. We present various results on hardness and hardness of approximation for this variant, by graph and district types.

Problem

• graph packing
• compact subgraphs
• districting
• balanced weight
• minimum weight threshold

Innovation

• graph packing
• balanced districts
• approximation algorithms
• minor-free graphs
• bounded expansion