Locally Fair Partitioning

TL;DR

This paper studies a novel local fairness concept in partitioning points into regions, focusing on 1D cases, characterizing when such partitions exist, and providing algorithms for their computation.

Contribution

It introduces the concept of local fairness in partitioning, analyzes existence conditions in 1D, and develops polynomial algorithms for computing fair partitions.

Findings

Locally fair partitions may not always exist depending on input parameters.

Approximate solutions are possible for clustered inputs with some size adjustments.

A polynomial-time algorithm is provided for computing locally fair partitions when they exist.

Abstract

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 $\Pi$ of the plane into regions so that each region contains roughly $\sigma = n/k$ points. $\Pi$ 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 $\Pi$ if it belongs to the minority party. A group $D$ of roughly $\sigma$ contiguous points is called a deviating group with respect to $\Pi$ if majority of points in $D$ are unhappy in $\Pi$. The partition $\Pi$ is locally fair if there is no deviating group with respect to $\Pi$. This paper focuses on a restricted case when points lie in $1$D. 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 $\sigma$, 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.