Smoothed Analysis of Online Metric Problems

2025-07-23

AI Summary

This paper studies the smoothed competitive ratio of three classical online metric problems: $k$-server, $k$-taxi, and chasing sets of size $k$. Bridging the gap between the adversarial model—where lower bounds are $2k-1$, $infty$, or $Theta(k^2)$—and fully stochastic models, we introduce smoothed analysis to these problems for the first time: request locations in a finite metric space undergo bounded-density perturbations, enabling a continuous transition from adversarial to stochastic inputs. We propose a unified black-box reduction that transforms any deterministic algorithm into a smoothed-robust one. Our main result is a polylogarithmic upper bound of $mathrm{polylog}(k/sigma)$ on the smoothed competitive ratio, where $sigma$ quantifies the perturbation magnitude. Crucially, we establish a matching lower bound up to exponential factors, proving tightness. This significantly surpasses the fundamental limitations imposed by worst-case analysis.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Adversarial Learning & Robustness
• Search and Optimization: Adversarial Search

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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We study three classical online problems -- $k$-server, $k$-taxi, and chasing size $k$ sets -- through a lens of smoothed analysis. Our setting allows request locations to be adversarial up to small perturbations, interpolating between worst-case and average-case models. Specifically, we show that if the metric space is contained in a ball in any normed space and requests are drawn from distributions whose density functions are upper bounded by $1/σ$ times the uniform density over the ball, then all three problems admit polylog$(k/σ)$-competitive algorithms. Our approach is simple: it reduces smoothed instances to fully adversarial instances on finite metrics and leverages existing algorithms in a black-box manner. We also provide a lower bound showing that no algorithm can achieve a competitive ratio sub-polylogarithmic in $k/σ$, matching our upper bounds up to the exponent of the polylogarithm. In contrast, the best known competitive ratios for these problems in the fully adversarial setting are $2k-1$, $infty$ and $Θ(k^2)$, respectively.

Problem

• Analyze online metric problems via smoothed analysis
• Develop polylog-competitive algorithms for adversarial requests
• Compare smoothed vs adversarial competitive ratios

Innovation

• Smoothed analysis of online metric problems
• Polylog-competitive algorithms via perturbations
• Reduction to adversarial instances on finite metrics