Locally Fair Partitioning

We model the societal task of redistricting political districts as a partitioning problem: Given set n points in plane, each belonging to one two parties, and parameter k, our goal is compute partition P plane into regions so that region contains roughly s = n/k points. should satisfy notion "local" fairness, which related core, well-studied concept cooperative game theory. A associated with majority party region, point unhappy if it belongs minority party. group D contiguous called deviating respect are P. The locally fair there no This paper focuses on restricted case when lie 1D. problem non-trivial even this case. consider both adversarial "beyond worst-case" settings for problem. For former, we characterize input parameters always exists; also show may not exist certain parameters. then models where "runs" red blue such clustered inputs, values s, but an approximate exists allow some have smaller sizes. finally present polynomial-time algorithm computing exists.