A $${o}\mathopen {}\left( n\right) \mathclose {}$$-Competitive Deterministic Algorithm for Online Matching a Line

Online matching on a line involves matching an online stream of requests to a given set of servers, all in the real line, with the objective of minimizing the sum of the distances between matched server-request pairs. The best previously known upper and lower bounds on the optimal deterministic competitive ratio are linear in the number of requests, and constant, respectively. We show that online matching on a line is essentially equivalent to a particular search problem, which we call k-lost-cows. We then obtain the first deterministic sub-linearly competitive algorithm for online matching on a line by giving such an algorithm for the k-lost-cows problem.