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 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).

Temporal Fairness in Multiwinner Voting

“…In contrast, in the scheduling model of Elkind, Kraiczy, and Teh (2022) or Bulteau et al (2021), timeslots can be rearranged arbitrarily, i.e., this model is fundamentally non-sequential. While nonsequential settings have interesting computational problems in their own right (Boehmer and Niedermeier 2021), sequential settings pose additional challenges.…”

“…For instance, one could mandate that there should not be κ timesteps that have elapsed such that an agent has not received a utility of at least γ. This idea was briefly mentioned by Boehmer and Niedermeier (2021) in a different, more abstract social choice scenario. Several other problems arise as well: e.g., one could ask what combinations of γ and κ can be accomplished by a specific voting rule (in the worst case, or on a specific instance), or by all rules satisfying a given set of axioms; one could also ask what is the minimum attainable γ for some fixed κ.…”

Maximizing Submodular Functions for Recommendation in the Presence of Biases

“…These variants extend the constant-factor approximation guarantee to other types of constraints (including upper bounds stated in Section 1), improve its running time, and design distributed variants of the algorithm [53,55,15,56]. Finally, motivated by context-specific fairness requirements, a number of recent works [17,4,3,11,10,68] also study submodular maximization in the presence of constraints beyond the family of constraints introduced in Section 1. However, unlike the present work, these works, assume that one can evaluate the true function F which may not be possible in the presence of biases.…”