A refined complexity analysis of fair districting over graphs

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 most frequent color occurs at given number of times more often than second color. is motivated various real-world scenarios where agents different types, which are one-to-one represented nodes network, have be partitioned disjoint districts. Herein, one strives for “fair districts” without any type being dominating majority This e.g. prevent segregation or political domination some party. conduct fine-grained analysis (parameterized) computational complexity . In particular, we prove it polynomial-time solvable on paths, cycles, stars, and caterpillars, but already becomes trees. Motivated latter negative result, perform parameterized with respect parameters including treewidth, problem-specific parameters, including, numbers colors obtain rich diverse, close complete picture corresponding landscape (that is, classification along classes FPT, XP, W[1]-hard, para-NP-hard).