Individual Fairness in Graph Decomposition

TL;DR

This paper introduces new algorithms for graph decomposition that ensure individual fairness by making the separation probabilities of node pairs at similar distances comparable, addressing fairness issues in classic methods.

Contribution

The paper presents novel algorithms that incorporate individual fairness into low diameter graph decompositions, balancing fairness with connectivity and cluster optimality.

Findings

Classic methods fail to ensure individual fairness.

New algorithms achieve fairness trade-offs with connectivity and cluster count.

Effective on real-world planar networks like congressional redistricting.

Abstract

In this paper, we consider classic randomized low diameter decomposition procedures for planar graphs that obtain connected clusters which are cohesive in that close-by pairs of nodes are assigned to the same cluster with high probability. We require the additional aspect of individual fairness - pairs of nodes at comparable distances should be separated with comparable probability. We show that classic decomposition procedures do not satisfy this property. We present novel algorithms that achieve various trade-offs between this property and additional desiderata of connectivity of the clusters and optimality in the number of clusters. We show that our individual fairness bounds may be difficult to improve by tying the improvement to resolving a major open question in metric embeddings. We finally show the efficacy of our algorithms on real planar networks modeling congressional redistricting.