Why Districting Becomes NP-hard

TL;DR

This work addresses edge-based districting by formalizing an exact mixed-integer programming framework for partitioning edges into $p$ districts centered at nodes, under balance, contiguity, and compactness criteria. It provides a complete subproblem complexity classification across all combinations of six design-criteria, proving polynomial-time solvability for several families (notably when excluding balance or keeping centers fixed) and NP-hardness for others via reductions from Vertex Cover and 3-Partition. The authors introduce a modular notation $\theta_{X}$ to denote subproblems and show how relaxing or adding criteria shifts complexity, with key results demonstrating that contiguity alone does not guarantee tractability and that balance interactions with center and connectivity can drive hardness. They also derive parameterized insights on balance, showing feasibility bounds for two-district cases using a tree-based construction and tight additive/multiplicative tolerances. Finally, the paper discusses implications for practice, including guidance on which constraints to relax for tractability and directions for future work on approximation algorithms for the hard variants, with extensions to node-based districting discussed throughout.

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.

Figures (7)

• Figure 1: The left graph violates vertex-induced subgraph connectivity as the darker vertices are not connected; the right graph satisfies edge-based connectivity for assignments as there is an allocation of edges such that all vertices of the same type are connected to each other.
• Figure 2: Example of a rounding graph and identified substructures for eliminating fractional variables
• Figure 3: A transformation from the 3-Partition problem of $S=\{4,5,5,5,5,6\}$ to $\theta_{B,C,I}$; the colors show a solution which directly correspond to the disjunction $S_1=\{4,5,6\}$ and $S_2=\{5,5,5\}$.
• Figure 4: A transformation from the 3-Partition problem of $S=\{4,5,5,5,5,6\}$ to $\theta_{B,I,W}$. The colors indicate the partition: dashed purple edges correspond to one subset ($S_1=\{4,5,6\}$), solid green edges to the other subset ($S_2=\{5,5,5\}$).
• Figure 5: Any optimal solution would not assign an edge to either the red or the blue center.

Theorems & Definitions (11)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Theorem 9
• Lemma 10