Minimizing Margin of Victory for Fair Political and Educational Districting

In many practical scenarios, a population is divided into disjoint groups for better administration, e.g., electorates into political districts, employees into departments, students into school districts, and so on. However, grouping people arbitrarily may lead to biased partitions, raising concerns of gerrymandering in political districting, racial segregation in schools, etc. To counter such issues, in this paper, we conceptualize such problems in a voting scenario, and propose FAIR DISTRICTING problem to divide a given set of people having preference over candidates into k groups such that the maximum margin of victory of any group is minimized. We also propose the FAIR CONNECTED DISTRICTING problem which additionally requires each group to be connected. We show that the FAIR DISTRICTING problem is NP-complete for plurality voting even if we have only 3 candidates but admits polynomial time algorithms if we assume k to be some constant or everyone can be moved to any group. In contrast, we show that the FAIR CONNECTED DISTRICTING problem is NP-complete for plurality voting even if we have only 2 candidates and k = 2. Finally, we propose heuristic algorithms for both the problems and show their effectiveness in UK political districting and in lowering racial segregation in public schools in the US.