Individual Preference Stability for Clustering

TL;DR

This paper introduces a new concept of individual preference stability in clustering, explores its computational complexity, and provides efficient algorithms for specific metric spaces, with empirical evaluation on real datasets.

Contribution

It defines IP-stability for clustering, proves NP-hardness in general, and offers polynomial-time algorithms for line and tree metrics, along with relaxed stability variants.

Findings

Exact IP-stability is NP-hard to decide in general.

Polynomial-time algorithms exist for line and tree metric spaces.

Relaxed stability algorithms provide practical clustering solutions.

Abstract

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.