$k$-Clustering via Iterative Randomized Rounding

2026-04-07

AI Summary

This work addresses the k-clustering problem with the constraint of opening exactly k facilities, focusing on approximation algorithms where the cost is defined as the p-th power of distances. Building upon the standard LP relaxation, the authors propose an iterative randomized rounding algorithm that, in expectation, satisfies the cardinality constraint while controlling the expected cost. The key contribution is the first LP-rounding-based (3^p + 1)/2–LMP (Lagrangian Multiplier Preserving) approximation algorithm for general L_p^p costs, unifying the treatment across different values of p. Using probabilistic rounding techniques, this LMP guarantee is converted into a true (1+ε)-approximation. Notably, the framework recovers the optimal (2+ε)-approximation for k-median, improves the metric k-means approximation ratio from 5.83 to 5+ε, and achieves a (4+ε)-approximation in Euclidean space matching the current best bound.

Abstract

In this work we propose a single rounding algorithm for the fractional solutions of the standard LP relaxation for $k$-clustering. As a starting point, we obtain an iterative rounding $(\frac{3^p + 1}{2})$-Lagrangian Multiplier-Perserving (LMP) approximation for the $k$-clustering problem with the cost function being the $p$-th power of the distance. Such an algorithm outputs a random solution that opens $k$ facilities \emph{in expectation}, whose cost in expectation is at most $\frac{3^p + 1}{2}$ times the optimum cost. Thus, we recover the $2$-LMP approximation for $k$-median by Jain et al.~[JACM'03], which played a central role in deriving the current best $2$ approximation for $k$-median. Unlike the result of Jain et al., our algorithm is based on LP rounding, and it can be easily adapted to the $L_p^p$-cost setting. For the Euclidean $k$-means problem, the LMP factor we obtain is $\frac{11}{3}$, which is better than the $5$ approximation given by this framework for general metrics. Then, we show how to convert the LMP-approximation algorithms to a true-approximation, with only a $(1+\varepsilon)$ factor loss in the approximation ratio. We obtain a ($\frac{3^p + 1}{2}+\varepsilon$)-approximation algorithm for $k$-clustering with cost function being the $p$-th power of the distance, for $p \geq 1$. This reproduces the best known ($2+\varepsilon$)-approximation for $k$-median by Cohen-Addad et al. [STOC'25], and improves the approximation factor for metric $k$-means from 5.83 by Charikar at al. [FOCS'25] to $5+\varepsilon$ in our framework. Moreover, the same algorithm, but with a specialized analysis, attains ($4+\varepsilon$)-approximation for Euclidean $k$-means matching the recent result by Charikar et al. [STOC'26].

Problem

• k-clustering
• approximation algorithm
• Lagrangian Multiplier-Preserving
• LP rounding
• k-means

Innovation

• iterative randomized rounding
• Lagrangian Multiplier-Preserving (LMP)
• k-clustering
• LP relaxation
• approximation algorithm