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1. Near-optimal Algorithms for Explainable k-Medians and k-Means

   Makarychev, Konstantin; Shan, Liren (July 2021, International Conference on Machine Learning)

   null (Ed.)

   We consider the problem of explainable k-medians and k-means introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). In this problem, our goal is to find a threshold decision tree that partitions data into k clusters and minimizes the k-medians or k-means objective. The obtained clustering is easy to interpret because every decision node of a threshold tree splits data based on a single feature into two groups. We propose a new algorithm for this problem which is O(log k) competitive with k-medians with ℓ1 norm and O(k) competitive with k-means. This is an improvement over the previous guarantees of O(k) and O(k^2) by Dasgupta et al (2020). We also provide a new algorithm which is O(log^{3}{2}k) competitive for k-medians with ℓ2 norm. Our first algorithm is near-optimal: Dasgupta et al (2020) showed a lower bound of Ω(log k) for k-medians; in this work, we prove a lower bound of Ω(k) for k-means. We also provide a lower bound of Ω(log k) for k-medians with ℓ2 norm. 

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2. Fast Optimal Circular Clustering and Applications on Round Genomes

   https://doi.org/10.1109/TCBB.2021.3077573  

   Debnath, Tathagata; Song, Mingzhou (June 2021, IEEE/ACM Transactions on Computational Biology and Bioinformatics)

   null (Ed.)

   Round genomes are found in bacteria, plant chloroplasts, and mitochondria. Genetic or epigenetic marks can present biologically interesting clusters along a circular genome. The circular data clustering problem groups N points on a circle into K clusters to minimize the within-cluster sum of squared distances. Repeatedly applying the K-means algorithm takes quadratic time, impractical for large circular datasets. To overcome this issue, we developed a fast, reproducible, and optimal circular clustering (FOCC) algorithm of worst-case O(KN log^2 N) time. The core is a fast optimal framed clustering algorithm, which we designed by integrating two divide-and-conquer and one bracket dynamic programming strategies. The algorithm is optimal based on a property of monotonic increasing cluster borders over frames on linearized data. On clustering 50,000 circular data points, FOCC outruns brute-force or heuristic circular clustering by three orders of magnitude. We produced clusters of CpG sites and genes along three round genomes, exhibiting higher quality than heuristic clustering. More broadly, the presented subquadratic-time algorithms offer the fastest known solution to not only framed and circular clustering, but also angular, periodical, and looped clustering. We implemented these algorithms in the R package OptCirClust (https://CRAN.R-project.org/package=OptCirClust) 

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3. Polylogarithmic Sketches for Clustering

   https://doi.org/10.4230/LIPIcs.ICALP.2022.38  

   Charikar, Moses; Waingarten, Erik (January 2022, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bojańczyk, Mikołaj; Merelli, Emanuela; Woodruff, David P (Ed.)

   Given n points in 𝓁_p^d, we consider the problem of partitioning points into k clusters with associated centers. The cost of a clustering is the sum of p-th powers of distances of points to their cluster centers. For p ∈ [1,2], we design sketches of size poly(log(nd),k,1/ε) such that the cost of the optimal clustering can be estimated to within factor 1+ε, despite the fact that the compressed representation does not contain enough information to recover the cluster centers or the partition into clusters. This leads to a streaming algorithm for estimating the clustering cost with space poly(log(nd),k,1/ε). We also obtain a distributed memory algorithm, where the n points are arbitrarily partitioned amongst m machines, each of which sends information to a central party who then computes an approximation of the clustering cost. Prior to this work, no such streaming or distributed-memory algorithm was known with sublinear dependence on d for p ∈ [1,2). 

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4. Random Cuts are Optimal for Explainable k-Medians

   Makarychev, Konstantin; Shan, Liren (December 2023, NeurIPS 2023)

   We show that the RandomCoordinateCut algorithm gives the optimal competitive ratio for explainable k-medians in l1. The problem of explainable k-medians was introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian in 2020. Several groups of authors independently proposed a simple polynomial-time randomized algorithm for the problem and showed that this algorithm is O(log k loglog k) competitive. We provide a tight analysis of the algorithm and prove that its competitive ratio is upper bounded by 2ln k +2. This bound matches the Omega(log k) lower bound by Dasgupta et al (2020). 

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5. 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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