Smoothed Analysis of Online Metric Problems

TL;DR

This paper applies smoothed analysis to classical online problems, showing that small perturbations in request locations lead to significantly improved competitive ratios, bridging worst-case and average-case scenarios.

Contribution

It introduces a smoothed analysis framework for online metric problems and provides polylogarithmic competitive algorithms with matching lower bounds, improving upon worst-case ratios.

Findings

Polylog$(k/σ)$-competitive algorithms for all three problems.

Lower bounds match upper bounds up to polylogarithmic factors.

Significant improvement over worst-case competitive ratios.

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 $1/\sigma$ times the uniform density over the ball, then all three problems admit polylog$(k/\sigma)$-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 $k/\sigma$, 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 $\Theta(k^2)$, respectively.