We study the minimumcost metric perfect matching problem under online i.i.d arrivals. We are given a fixed metric with a server at each of the points, and then requests arrive online, each drawn independently from a known probability distribution over the points. Each request has to be matched to a free server, with cost equal to the distance. The goal is to minimize the expected total cost of the matching. Such stochastic arrival models have been widely studied for the maximization variants of the online matching problem; however, the only known result for the minimization problem is a tight O(log n)competitiveness for the randomorder arrival model. This is in contrast with the adversarial model, where an optimal competitive ratio of O(log n) has long been conjectured and remains a tantalizing open question. In this paper, we show that the i.i.d model admits substantially better algorithms: our main result is an O((log log log n)^2)competitive algorithm in this model, implying a strict separation between the i.i.d model and the adversarial and random order models. Along the way we give a 9competitive algorithm for the line and tree metrics  the first O(1)competitive algorithm for any nontrivial arrival model for these muchstudied metrics.
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Chasing Convex Bodies with Linear Competitive Ratio
We study the problem of chasing convex bodies online: given a sequence of convex bodies the algorithm must respond with points in an online fashion (i.e., is chosen before is revealed). The objective is to minimize the sum of distances between successive points in this sequence. Bubeck et al. (STOC 2019) gave a competitive algorithm for this problem. We give an algorithm that is competitive for any sequence of length .
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 NSFPAR ID:
 10327024
 Date Published:
 Journal Name:
 Journal of the ACM
 Volume:
 68
 Issue:
 5
 ISSN:
 00045411
 Page Range / eLocation ID:
 1 to 10
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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