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Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals

Authors Zhiyi Huang, Zhihao Gavin Tang, Xiaowei Wu, Yuhao Zhang

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Online algorithms
  • Theory of computation → Linear programming
Keywords
  • Vertex Weighted
  • Online Bipartite Matching
  • Randomized Primal-Dual

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Abstract

We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time goes by, and each online vertex matches the neighbor with the highest offer at its arrival.

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Zhiyi Huang, Zhihao Gavin Tang, Xiaowei Wu, and Yuhao Zhang. Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 79:1-79:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ICALP.2018.79

Author Details

Zhiyi Huang
  • Department of Computer Sicence, The University of Hong Kong, Hong Kong
Zhihao Gavin Tang
  • Department of Computer Sicence, The University of Hong Kong, Hong Kong
Xiaowei Wu
  • Department of Computing, The Hong Kong Polytechnic University, Hong Kong
Yuhao Zhang
  • Department of Computer Sicence, The University of Hong Kong, Hong Kong

Funding

  • Huang, Zhiyi: Partially supported by the Hong Kong RGC under the grant HKU17202115E.
  • Tang, Zhihao Gavin: Partially supported by his supervisor Hubert Chan’s Hong Kong RGC grant 17202715.
  • Wu, Xiaowei: Part of the work was done when the author was a postdoc at the University of Hong Kong.

References

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