Individual Preference Stability for Clustering
This addresses the need for fair and stable clustering in applications like game theory and algorithmic fairness, though it is incremental as it builds on existing clustering concepts with new constraints.
The paper tackles the problem of defining and computing clusterings that satisfy individual preference stability, where each point is on average closer to its own cluster than to others, showing NP-hardness in general but providing polynomial-time algorithms for specific metric spaces like the real line and tree metrics.
In this paper, we propose a natural notion of individual preference (IP) stability for clustering, which asks that every data point, on average, is closer to the points in its own cluster than to the points in any other cluster. Our notion can be motivated from several perspectives, including game theory and algorithmic fairness. We study several questions related to our proposed notion. We first show that deciding whether a given data set allows for an IP-stable clustering in general is NP-hard. As a result, we explore the design of efficient algorithms for finding IP-stable clusterings in some restricted metric spaces. We present a polytime algorithm to find a clustering satisfying exact IP-stability on the real line, and an efficient algorithm to find an IP-stable 2-clustering for a tree metric. We also consider relaxing the stability constraint, i.e., every data point should not be too far from its own cluster compared to any other cluster. For this case, we provide polytime algorithms with different guarantees. We evaluate some of our algorithms and several standard clustering approaches on real data sets.