A Refined Complexity Analysis of Fair Districting over Graphs

by   Niclas Boehmer, et al.

We study the NP-hard Fair Connected Districting problem: Partition a vertex-colored graph into k connected components (subsequently referred to as districts) so that in each 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. Our work builds on a model recently proposed by Stoica et al. [AAMAS 2020], thereby also strengthening and extending computational hardness results from there. More specifically, with Fair Connected Districting we identify a natural, already hard special case of their Fair Connected Regrouping problem. We conduct a fine-grained analysis of the (parameterized) computational complexity of Fair Connected Districting, proving that it is polynomial-time solvable on paths, cycles, stars, caterpillars, and cliques, 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]-hardness, and para-NP-hardness). Doing so, we draw a fine line between tractability and intractability and identify structural properties of the underlying graph that make Fair Connected Districting computationally hard.


The Complexity of Gerrymandering Over Graphs: Paths and Trees

Roughly speaking, gerrymandering is the systematic manipulation of the b...

Grid Recognition: Classical and Parameterized Computational Perspectives

Grid graphs, and, more generally, k× r grid graphs, form one of the most...

Computing Fair Utilitarian Allocations of Indivisible Goods

We study the computational complexity of computing allocations that are ...

On Parameterized Complexity of Liquid Democracy

In liquid democracy, each voter either votes herself or delegates her vo...

Minimizing Margin of Victory for Fair Political and Educational Districting

In many practical scenarios, a population is divided into disjoint group...

Multivariate Analysis of Scheduling Fair Competitions

A fair competition, based on the concept of envy-freeness, is a non-elim...

Fine-Grained View on Bribery for Group Identification

Given a set of agents qualifying or disqualifying each other, group iden...