Adaptive Sampling for Minimum-Norm $k$-Clustering

Haripriya Pulyassary; Chaitanya Swamy

Home/Archive/arXiv

arXiv · 2607.12421

Adaptive Sampling for Minimum-Norm $k$-Clustering

Haripriya Pulyassary·Chaitanya Swamy

View original source ↗Read source document ↗Export metadata ↓

Abstract

In $k$-clustering problems, we are given a metric space $(\mathcal{C}, d)$, and must choose a set $S$ of $k$ centers to open. Each client $j \in \mathcal{C}$ incurs an assignment cost, which is the distance between $j$ and center in $S$ that it has been assigned to. In this work, we study the \emph{minimum-norm $k$-clustering problem}, where we are given an arbitrary monotone symmetric norm $f$, and wish to open $k$ centers so as to minimize $f$(assignment-cost vector). This is a powerful generalization, encompassing many classical $k$-clustering problems including the $k$-median, $k$-means, and $k$-center problems. A simple and efficient algorithmic idea is that of \emph{adaptive sampling}, wherein we randomly choose the location of the next center to open with probability proportional to its ``cost" under the currently chosen set. While this has yielded fast algorithms for some $k$-clustering problem, little is known for settings \emph{without} ``min-sum" objectives. We devise the first adaptive-sampling-based bicriteria constant-factor approximation algorithm for general minimum-norm $k$-clustering, vastly expanding the scope of problems handled by adaptive sampling. For the special case of $\text{Top}_\ell$ norms, which form a building block of monotone symmetric norms, we show that adaptive sampling yields an $O(\log k)$-approximation algorithm.

Explore related subjects

cs.DS

Keep this discovery

BibTeX RIS

Haripriya Pulyassary, Chaitanya Swamy. 2026-07-14. Adaptive Sampling for Minimum-Norm $k$-Clustering. https://arxiv.org/abs/2607.12421

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗

arXiv2026

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

cs.DS↗

arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗