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Multiplicative auction algorithm for approximate maximum weight bipartite matching

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Abstract

We present an auction algorithm using multiplicative instead of constant weight updates to compute a \((1-\varepsilon )\)-approximate maximum weight matching (MWM) in a bipartite graph with n vertices and m edges in time \(O(m\varepsilon ^{-1})\), beating the running time of the fastest known approximation algorithm of Duan and Pettie [JACM ’14] that runs in \(O(m\varepsilon ^{-1}\log \varepsilon ^{-1})\). Our algorithm is very simple and it can be extended to give a dynamic data structure that maintains a \((1-\varepsilon )\)-approximate maximum weight matching under (1) one-sided vertex deletions (with incident edges) and (2) one-sided vertex insertions (with incident edges sorted by weight) to the other side. The total time time used is \(O(m\varepsilon ^{-1})\), where m is the sum of the number of initially existing and inserted edges.

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Notes

  1. By approximate solution we mean a possibly fractional assignments of variables that obtains an approximately good LP objective. If we find such an approximate solution to MWM, fractional solutions need to be rounded to obtain an actual matching.

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Acknowledgements

The first author thanks Chandra Chekuri for useful discussions about this paper.

Funding

Monika Henzinger: This work was done in part at the University of Vienna. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 101019564 “The Design of Modern Fully Dynamic Data Structures (MoDynStruct)” and from the Austrian Science Fund (FWF) project “Fast Algorithms for a Reactive Network Layer (ReactNet)”, P 33775-N, with additional funding from the netidee SCIENCE Stiftung, 2020–2024.

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  1. University of Illinois Urbana-Champaign, Urbana, IL, USA

    Da Wei Zheng

  2. Institute of Science and Technology Austria (ISTA), Klosterneuburg, Austria

    Monika Henzinger

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  1. Da Wei Zheng
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Correspondence to Da Wei Zheng or Monika Henzinger.

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An earlier version of this paper appeared in the 24th International Conference on Integer Programming and Combinatorial Optimization (IPCO 2023) with a slightly slower algorithm running in \(O(m\varepsilon ^{-1}\log \varepsilon ^{-1})\) time.

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Zheng, D.W., Henzinger, M. Multiplicative auction algorithm for approximate maximum weight bipartite matching. Math. Program. (2024). https://doi.org/10.1007/s10107-024-02066-3

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