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

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 [KP93, KMV94] that achieve a competitive ratio of 2k−1, and this bound is tight for deterministic algorithms. Randomization can be used to overcome this lower bound, and we know O(log 2 k) competitive algorithms [BBGN07] and an Ω(log k) lower bound. 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 [Rag18]. 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 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. We finally illustrate another benefit of allowing limited recourse: we can extend the Online Metric Matching model to handle arrivals and departures of both clients and servers (as opposed to just handling arrivals of clients) and still maintain competitive solutions. Indeed, we show a simple randomized O(log n)-competitive algorithm with O(log ∆)-recourse in this fully online setting, where n is the number of points in the metric space and ∆ is the aspect ratio of the underlying metric. Perhaps the most important technical contribution of this work is in showing that these improved results can, in fact, be achieved by suitably equipping a two-decades-old algorithm PERMUTATION for online metric matching with limited recourse.