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1. Locally Private k-Means Clustering with Constant Multiplicative Approximation and Near-Optimal Additive Error

   https://doi.org/10.1609/aaai.v36i6.20565  

   Chaturvedi, Anamay; Jones, Matthew; Nguyen, Huy L. (January 2022, Proceedings of the AAAI Conference on Artificial Intelligence)

   Given a data set of size n in d'-dimensional Euclidean space, the k-means problem asks for a set of k points (called centers) such that the sum of the l_2^2-distances between the data points and the set of centers is minimized. Previous work on this problem in the local differential privacy setting shows how to achieve multiplicative approximation factors arbitrarily close to optimal, but suffers high additive error. The additive error has also been seen to be an issue in implementations of differentially private k-means clustering algorithms in both the central and local settings. In this work, we introduce a new locally private k-means clustering algorithm that achieves near-optimal additive error whilst retaining constant multiplicative approximation factors and round complexity. Concretely, given any c>sqrt(2), our algorithm achieves O(k^(1 + O(1/(2c^2-1))) * sqrt(d' n) * log d' * poly log n) additive error with an O(c^2) multiplicative approximation factor. 

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2. Differentially Private k-Means via Exponential Mechanism and Max Cover

   Nguyen, H L; Chaturvedi, A; Xu, E Z. (January 2021, Proceedings of the AAAI Conference on Artificial Intelligence)

   We introduce a new (ϵₚ, δₚ)-differentially private algorithm for the k-means clustering problem. Given a dataset in Euclidean space, the k-means clustering problem requires one to find k points in that space such that the sum of squares of Euclidean distances between each data point and its closest respective point among the k returned is minimised. Although there exist privacy-preserving methods with good theoretical guarantees to solve this problem, in practice it is seen that it is the additive error which dictates the practical performance of these methods. By reducing the problem to a sequence of instances of maximum coverage on a grid, we are able to derive a new method that achieves lower additive error than previous works. For input datasets with cardinality n and diameter Δ, our algorithm has an O(Δ² (k log² n log(1/δₚ)/ϵₚ + k √(d log(1/δₚ))/ϵₚ)) additive error whilst maintaining constant multiplicative error. We conclude with some experiments and find an improvement over previously implemented work for this problem. 

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3. On Approximability of 𝓁₂² Min-Sum Clustering

   https://doi.org/10.4230/lipics.socg.2025.62  

   S, Karthik C; Lee, Euiwoong; Rabani, Yuval; Schwiegelshohn, Chris; Zhou, Samson (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Aichholzer, Oswin; Wang, Haitao (Ed.)

   The 𝓁₂² min-sum k-clustering problem is to partition an input set into clusters C_1,…,C_k to minimize ∑_{i=1}^k ∑_{p,q ∈ C_i} ‖p-q‖₂². Although 𝓁₂² min-sum k-clustering is NP-hard, it is not known whether it is NP-hard to approximate 𝓁₂² min-sum k-clustering beyond a certain factor. In this paper, we give the first hardness-of-approximation result for the 𝓁₂² min-sum k-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than 1.056 and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327. We then complement our hardness result by giving a fast PTAS for 𝓁₂² min-sum k-clustering. Specifically, our algorithm runs in time O(n^{1+o(1)}d⋅ 2^{(k/ε)^O(1)}), which is the first nearly linear time algorithm for this problem. We also consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label i ∈ [k] for input point, thereby implicitly partitioning the input dataset into k clusters that induce an approximately optimal solution, up to some amount of adversarial error α ∈ [0,1/2). We give a polynomial-time algorithm that outputs a (1+γα)/(1-α)²-approximation to 𝓁₂² min-sum k-clustering, for a fixed constant γ > 0. 

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4. A Framework for Parallelizing Hierarchical Clustering Methods

   Silvio Lattanzi, Thomas Lavastida (January 2019, European Conference on Machine Learning)

   Hierarchical clustering is a fundamental tool in data mining, machine learning and statistics. Popular hierarchical clustering algorithms include top-down divisive approaches such as bisecting k-means, k-median, and k-center and bottom-up agglomerative approaches such as single- linkage, average-linkage, and centroid-linkage. Unfortunately, only a few scalable hierarchical clustering algorithms are known, mostly based on the single-linkage algorithm. So, as datasets increase in size every day, there is a pressing need to scale other popular methods. We introduce efficient distributed algorithms for bisecting k-means, k- median, and k-center as well as centroid-linkage. In particular, we first formalize a notion of closeness for a hierarchical clustering algorithm, and then we use this notion to design new scalable distributed methods with strong worst case bounds on the running time and the quality of the solutions. Finally, we show experimentally that the introduced algorithms are efficient and close to their sequential variants in practice. 

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5. Improved Approximation Algorithms for Relational Clustering

   https://doi.org/10.1145/3695831  

   Esmailpour, Aryan; Sintos, Stavros (November 2024, Proceedings of the ACM on Management of Data)

   Clustering plays a crucial role in computer science, facilitating data analysis and problem-solving across numerous fields. By partitioning large datasets into meaningful groups, clustering reveals hidden structures and relationships within the data, aiding tasks such as unsupervised learning, classification, anomaly detection, and recommendation systems. Particularly in relational databases, where data is distributed across multiple tables, efficient clustering is essential yet challenging due to the computational complexity of joining tables. This paper addresses this challenge by introducing efficient algorithms for k-median and k-means clustering on relational data without the need for pre-computing the join query results. For the relational k-median clustering, we propose the first efficient relative approximation algorithm. For the relational k-means clustering, our algorithm significantly improves both the approximation factor and the running time of the known relational k-means clustering algorithms, which suffer either from large constant approximation factors, or expensive running time. Given a join query q and a database instance D of O(N) tuples, for both k-median and k-means clustering on the results of q on D, we propose randomized (1+ε)γ-approximation algorithms that run in roughly O(k²N^fhw)+T_γ(k²) time, where ε ∈ (0,1) is a constant parameter decided by the user, \fhw is the fractional hyper-tree width of Q, while γ and T_γ(x) represent the approximation factor and the running time, respectively, of a traditional clustering algorithm in the standard computational setting over x points. 

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