A $(2+varepsilon)$-Approximation Algorithm for Metric $k$-Median

2025-03-14

AI Summary

This paper studies the $k$-median problem in metric spaces: given $n$ clients and a set of candidate facility locations equipped with a distance function satisfying the triangle inequality, select exactly $k$ centers to minimize the total distance from each client to its nearest center. Addressing the long-standing $2.613$ lower bound on the approximability ratio, we present the first $(2+varepsilon)$-approximation algorithm (for any $varepsilon > 0$) with rigorous theoretical guarantees under stability assumptions. Our method integrates an adaptive multi-stage greedy framework, $k$-means++-inspired sampling, submodular optimization, and partition matroid constraints. A key technical innovation is the “walking solution path” analysis technique, enabling precise control over solution evolution. The algorithm uses at most $O(log n / varepsilon^2)$ extra centers beyond $k$, substantially improving the known approximation boundary for this classic NP-hard problem.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

In the classical NP-hard metric $k$-median problem, we are given a set of $n$ clients and centers with metric distances between them, along with an integer parameter $kgeq 1$. The objective is to select a subset of $k$ open centers that minimizes the total distance from each client to its closest open center. In their seminal work, Jain, Mahdian, Markakis, Saberi, and Vazirani presented the Greedy algorithm for facility location, which implies a $2$-approximation algorithm for $k$-median that opens $k$ centers in expectation. Since then, substantial research has aimed at narrowing the gap between their algorithm and the best achievable approximation by an algorithm guaranteed to open exactly $k$ centers. During the last decade, all improvements have been achieved by leveraging their algorithm or a small improvement thereof, followed by a second step called bi-point rounding, which inherently increases the approximation guarantee. Our main result closes this gap: for any $epsilon>0$, we present a $(2+epsilon)$-approximation algorithm for $k$-median, improving the previous best-known approximation factor of $2.613$. Our approach builds on a combination of two algorithms. First, we present a non-trivial modification of the Greedy algorithm that operates with $O(log n/epsilon^2)$ adaptive phases. Through a novel walk-between-solutions approach, this enables us to construct a $(2+epsilon)$-approximation algorithm for $k$-median that consistently opens at most $k + O(log n{/epsilon^2})$ centers. Second, we develop a novel $(2+epsilon)$-approximation algorithm tailored for stable instances, where removing any center from an optimal solution increases the cost by at least an $Omega(epsilon^3/log n)$ fraction. Achieving this involves a sampling approach inspired by the $k$-means++ algorithm and a reduction to submodular optimization subject to a partition matroid.

Problem

• Develops a (2+ε)-approximation algorithm for the NP-hard metric k-median problem.
• Improves the approximation factor from 2.613 to 2+ε for k-median.
• Introduces a novel walk-between-solutions approach and a stable instance algorithm.

Innovation

• Modified Greedy algorithm with adaptive phases
• Walk-between-solutions approach for approximation
• Sampling and submodular optimization for stable instances