A Refined Complexity Analysis of Fair Districting over Graphs

TL;DR

This paper provides a detailed complexity analysis of the NP-hard Fair Connected Districting problem, identifying cases where it is polynomial-time solvable and establishing its parameterized complexity landscape.

Contribution

It offers a refined, parameterized complexity classification of the Fair Connected Districting problem across various graph classes and parameters.

Findings

Polynomial-time solvable on paths, cycles, stars, and caterpillars.

NP-hard on trees.

Complete parameterized complexity landscape with classifications like FPT, XP, W[1]-hard.

Abstract

We study the NP-hard Fair Connected Districting problem recently proposed by Stoica et al. [AAMAS 2020]: Partition a vertex-colored graph into k connected components (subsequently referred to as districts) so that in every district the most frequent color occurs at most a given number of times more often than the second most frequent color. Fair Connected Districting is motivated by various real-world scenarios where agents of different types, which are one-to-one represented by nodes in a network, have to be partitioned into disjoint districts. Herein, one strives for "fair districts" without any type being in a dominating majority in any of the districts. This is to e.g. prevent segregation or political domination of some political party. We conduct a fine-grained analysis of the (parameterized) computational complexity of Fair Connected Districting. In particular, we prove that it is polynomial-time solvable on paths, cycles, stars, and caterpillars, but already becomes NP-hard on trees. Motivated by the latter negative result, we perform a parameterized complexity analysis with respect to various graph parameters, including treewidth, and problem-specific parameters, including, the numbers of colors and districts. We obtain a rich and diverse, close to complete picture of the corresponding parameterized complexity landscape (that is, a classification along the complexity classes FPT, XP, W[1]-hard, and para-NP-hard).