Online bipartite matching with amortized O(log2 n) replacements

Aaron Bernstein, Jacob Holm, Eva Rotenberg

Research output: Chapter in Book/Report/Conference proceedingArticle in proceedingsResearchpeer-review


In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one by one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized <i>O</i>(log<sup>2</sup> <i>n</i>) replacements per insertion, where <i>n</i> is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for <i>any</i> replacement strategy, almost matching the Ω(log <i>n</i>) lower bound. The previous best strategy known achieved amortized [EQUATION] replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than then trivial <i>O</i>(<i>n</i>) bound was known except in special cases. Our analysis immediately implies the same upper bound of <i>O</i>(log<sup>2</sup> <i>n</i>) reassignments for the capacitated assignment problem, where each vertex on the static side of the bipartition is initialized with the capacity to serve a number of vertices. We also analyze the problem of minimizing the maximum server load. We show that if the final graph has maximum server load <i>L</i>, then the SAP protocol makes amortized <i>O</i>(min{<i>L</i> log<sup>2</sup> <i>n</i>, [EQUATION] reassignments. We also show that this is close to tight because Ω(min{<i>L</i>, [EQUATION] reassignments can be necessary.
Original languageEnglish
Title of host publicationProceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms
PublisherSociety for Industrial and Applied Mathematics
Publication date2018
ISBN (Electronic)978-1-61197-503-1
Publication statusPublished - 2018
Event29th Annual ACM-SIAM Symposium on Discrete Algorithms - Astor Crowne Plaze, New Orleans French Quarter , New Orleans , United States
Duration: 7 Jan 201810 Jan 2018
Conference number: 29


Conference29th Annual ACM-SIAM Symposium on Discrete Algorithms
LocationAstor Crowne Plaze, New Orleans French Quarter
CountryUnited States
CityNew Orleans
SeriesProceedings of the Twenty-ninth Annual Acm-siam Symposium


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