PERMUTATION Strikes Back: The Power of Recourse in Online Metric Matching

TL;DR

This paper demonstrates that allowing a small amount of recourse in online metric matching significantly improves competitive ratios, achieving near-optimal solutions in general and line metrics with minimal adjustments.

Contribution

It introduces the first deterministic algorithms with recourse that surpass classical lower bounds for online metric matching in general and line metrics.

Findings

Deterministic $O( ext{log }k)$-competitive algorithm with $O( ext{log }k)$ recourse for general metrics.

Deterministic 3-competitive algorithm with $O( ext{log }k)$ recourse for line metrics.

Recourse greatly enhances the quality of online matchings under minimal modifications.

Abstract

In the classical Online Metric Matching problem, we are given a metric space with $k$ servers. A collection of clients arrive in an online fashion, and upon arrival, a client should irrevocably be matched to an as-yet-unmatched server. The goal is to find an online matching which minimizes the total cost, i.e., the sum of distances between each client and the server it is matched to. We know deterministic algorithms~\cite{KP93,khuller1994line} that achieve a competitive ratio of $2k-1$, and this bound is tight for deterministic algorithms. The problem has also long been considered in specialized metrics such as the line metric or metrics of bounded doubling dimension, with the current best result on a line metric being a deterministic $O(\log k)$ competitive algorithm~\cite{raghvendra2018optimal}. Obtaining (or refuting) $O(\log k)$-competitive algorithms in general metrics and constant-competitive algorithms on the line metric have been long-standing open questions in this area. In this paper, we investigate the robustness of these lower bounds by considering the Online Metric Matching with Recourse problem where we are allowed to change a small number of previous assignments upon arrival of a new client. Indeed, we show that a small logarithmic amount of recourse can significantly improve the quality of matchings we can maintain. For general metrics, we show a simple \emph{deterministic} $O(\log k)$-competitive algorithm with $O(\log k)$-amortized recourse, an exponential improvement over the $2k-1$ lower bound when no recourse is allowed. We next consider the line metric, and present a deterministic algorithm which is $3$-competitive and has $O(\log k)$-recourse, again a substantial improvement over the best known $O(\log k)$-competitive algorithm when no recourse is allowed.