A Refined Complexity Analysis of Fair Districting over Graphs

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 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, such as district-based elections, where agents of different types, which are one-to-one represented by nodes in a network, have to be partitioned into disjoint districts. We conduct a fine-grained analysis of the (parameterized) computational complexity of Fair Connected Districting: We study its parameterized complexity with respect to various graph parameters, including treewidth, and problem-specific parameters, including the numbers of colors and districts, and its complexity on graphs from different classes (such as paths, stars, and trees).