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