Online bipartite matching with amortized O(log2 n) replacements

Aaron Bernstein, Jacob Holm, Eva Rotenberg

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    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 O(log2 n) replacements per insertion, where n is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for any replacement strategy, almost matching the (log n) lower bound. The previous best strategy known achieved amortized O(p n) replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than then trivial O(n) bound was known except in special cases. Our analysis immediately implies the same upper bound of O(log2 n) 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 L, then the SAP protocol makes amortized O(min-Llog2 n, p n log n}) reassignments. We also show that this is close to tight because (min-L, p n}) reassignments can be necessary.

    Original languageEnglish (US)
    Title of host publication29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
    EditorsArtur Czumaj
    PublisherAssociation for Computing Machinery
    Pages947-959
    Number of pages13
    ISBN (Electronic)9781611975031
    DOIs
    StatePublished - 2018
    Event29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 - New Orleans, United States
    Duration: Jan 7 2018Jan 10 2018

    Publication series

    NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

    Other

    Other29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018
    Country/TerritoryUnited States
    CityNew Orleans
    Period1/7/181/10/18

    ASJC Scopus subject areas

    • Software
    • General Mathematics

    Fingerprint

    Dive into the research topics of 'Online bipartite matching with amortized O(log2 n) replacements'. Together they form a unique fingerprint.

    Cite this