Why Districting Becomes NP-hard

TL;DR

This paper analyzes the computational complexity of districting problems, identifying which constraints lead to NP-hardness and which subproblems are tractable, providing insights into the problem's inherent difficulty.

Contribution

It precisely characterizes the boundary between tractable and NP-hard districting subproblems based on constraint relaxation and removal.

Findings

Certain constraint groups make districting NP-hard

Relaxing specific constraints can lead to polynomial-time solutions

Implications for node-based districting problems and alternative criteria

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.