Randomizing the Number of Centers in k-means++

Fairness in clustering has attracted sustained research interest, motivated by the need to ensure equitable representation of protected groups in machine learning applications. We study fair $k$-means clustering in Euclidean space, where the proportion of each protected group in every cluster must lie within specified lower and upper bounds. These constraints make it challenging to determine both cluster centers and point assignments. We propose an approximation algorithm that combines a linear programming relaxation with geometric transformations of the input to construct candidate center sets. Given a $ρ$-approximate algorithm for weighted $k$-means and any $ε>0$, our algorithm returns a fractional solution whose cost is at most $1+(3-1/Γ)ρ+O(ε)$ times the optimal integral fair cost, where $Γ\approx6.357$ is an upper bound on the integrality gap of the standard Euclidean $k$-means LP. With a PTAS as the subroutine, the approximation ratio becomes $3.8427+O(ε)$, improving the previous factor of $5+O(ε)$ to below $4$. The solution satisfies all fairness constraints exactly and can be rounded to an integral assignment with a bounded additive violation of fairness and no increase in cost. The same approximation guarantee extends to the $k$-sparse Wasserstein barycenter problem.

Clustering problems such as $k$-means and $k$-median are staples of unsupervised learning, and many algorithmic techniques have been developed to tackle their numerous aspects. In this paper, we focus on the class of greedy approximation algorithm, that attracted less attention than local-search or primal-dual counterparts. In particular, we study the recursive greedy algorithm developed by Mettu and Plaxton [SIAM J. Comp 2003]. We provide a simplification of the algorithm, allowing for faster implementation, in graph metrics or in Euclidean space, where our algorithm matches or improves the state-of-the-art.

The k-means++ algorithm is commonly restarted multiple times to avoid poor local optima, yet the number of restarts is almost always chosen arbitrarily and applied uniformly regardless of data set difficulty. This undermines any comparison relying on such a choice and wastes computation on easy data sets while potentially under-serving hard ones. We introduce GTRC, a restart criterion combining a Good-Turing estimate, a proven unconditional bound, and a confidence-based bound on the probability that a further restart would improve on the current result, stopping once this probability falls below a user-specified tolerance $\varepsilon$. Across 36 data sets, GTRC reached clustering quality competitive with well-chosen fixed restart counts, while the number of restarts used varied considerably and appropriately with data set difficulty, governed by an interpretable, data-dependent signal rather than a fixed rule. GTRC offers a principled and reportable alternative to fixing the number of $k$-means++ restarts in advance. Software:https://github.com/RCdeAmorim/Good-Turing-Restart-Criterion.