Why Districting Becomes NP-hard

2025-10-29

AI Summary

This paper investigates the computational complexity boundary of edge-based districting: it precisely characterizes the phase transition—from polynomial-time solvability (P) to NP-hardness—induced by relaxing or removing constraints such as balance, connectivity, and compactness. Using an exact integer programming formulation and rigorous computational complexity analysis, the work establishes necessary and sufficient conditions for tractability under all combinations of these constraints, identifying connectivity as the critical source of NP-hardness. The results are extended to node-based partitioning and multiple fairness criteria variants. Collectively, this work constructs the first theoretical tractability map for districting problems, providing tight complexity-theoretic bounds and actionable design principles for developing approximation algorithms and practical redistricting tools.

Technology Category

Constraint Satisfaction and Optimization: Distributed CSP/OptimizationKnowledge Representation and Reasoning: Computational Complexity of ReasoningSearch and Optimization: Combinatorial Optimization

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsEconomics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environmentsSocial Networks and Social Media: Fairness and bias in social network and social media analysis

Abstract

This paper investigates why and when the edge-based districting problem becomes computationally intractable. The overall problem is represented as an exact mathematical programming formulation consisting of an objective function and several constraint groups, each enforcing a well-known districting criterion such as balance, contiguity, or compactness. While districting is known to be NP-hard in general, we study what happens when specific constraint groups are relaxed or removed. The results identify precise boundaries between tractable subproblems (in P) and intractable ones (NP-hard). The paper also discusses implications on node-based analogs of the featured districting problems, and it considers alternative notions of certain criteria in its analysis.

Problem

• Investigates computational intractability boundaries in edge-based districting
• Identifies tractable versus NP-hard subproblems through constraint relaxation
• Analyzes implications for node-based districting and alternative criteria

Innovation

• Exact mathematical programming formulation for districting
• Relaxing constraints to identify tractability boundaries
• Analyzing NP-hard transitions in edge-based districting