{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article9058","name":"A Robust and Optimal Online Algorithm for Minimum Metric Bipartite Matching","abstract":"We study the Online Minimum Metric Bipartite Matching Problem. In this problem, we are given point sets S and R which correspond to the server and request locations; here |S|=|R|=n. All these locations are points from some metric space and the cost of matching a server to a request is given by the distance between their locations in this space. In this problem, the request points arrive one at a time. When a request arrives, we must immediately and irrevocably match it to a \"free\" server. The matching obtained after all the requests are processed is the online matching M. The cost of M is the sum of the cost of its edges. The performance of any online algorithm is the worst-case ratio of the cost of its online solution M to the minimum-cost matching.\r\n\r\nWe present a deterministic online algorithm for this problem. Our algorithm is the first to simultaneously achieve optimal performances in the well-known adversarial and the random arrival models. For the adversarial model, we obtain a competitive ratio of 2n-1 + o(1); it is known that no deterministic algorithm can do better than 2n-1. In the random arrival model, our algorithm obtains a competitive ratio of 2H_n - 1 + o(1); where H_n is the n-th Harmonic number. We also prove that any online algorithm will have a competitive ratio of at least 2H_n - 1-o(1) in this model.\r\n\r\nWe use a new variation of the offline primal-dual method for computing minimum cost matching to compute the online matching. Our primal-dual method is based on a relaxed linear-program. Under metric costs, this specific relaxation helps us relate the cost of the online matching with the offline matching leading to its robust properties.","keywords":["Online Algorithms","Metric Bipartite Matching"],"author":{"@type":"Person","name":"Raghvendra, Sharath","givenName":"Sharath","familyName":"Raghvendra"},"position":18,"pageStart":"18:1","pageEnd":"18:16","dateCreated":"2016-09-06","datePublished":"2016-09-06","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Raghvendra, Sharath","givenName":"Sharath","familyName":"Raghvendra"},"copyrightYear":"2016","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.APPROX-RANDOM.2016.18","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":"http:\/\/dx.doi.org\/10.1145\/2213977.2214014","isPartOf":{"@type":"PublicationVolume","@id":"#volume6263","volumeNumber":60,"name":"Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX\/RANDOM 2016)","dateCreated":"2016-09-06","datePublished":"2016-09-06","editor":[{"@type":"Person","name":"Jansen, Klaus","givenName":"Klaus","familyName":"Jansen"},{"@type":"Person","name":"Mathieu, Claire","givenName":"Claire","familyName":"Mathieu"},{"@type":"Person","name":"Rolim, Jos\u00e9 D. P.","givenName":"Jos\u00e9 D. P.","familyName":"Rolim"},{"@type":"Person","name":"Umans, Chris","givenName":"Chris","familyName":"Umans"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article9058","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6263"}}}