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International Workshop on Approximation and Online Algorithms

WAOA 2017: Approximation and Online Algorithms pp 253–268Cite as

On Conceptually Simple Algorithms for Variants of Online Bipartite Matching

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10787))

Abstract

We present a series of results regarding conceptually simple algorithms for bipartite matching in various online and related models. We first consider a deterministic adversarial model. The best approximation ratio possible for a single-pass deterministic online algorithm is 1/2, which is achieved by any greedy algorithm. Dürr et al. [15] recently presented a 2-pass algorithm called Category-Advice that achieves approximation ratio 3/5. We extend their algorithm to multiple passes. We prove the exact approximation ratio for the k-pass Category-Advice algorithm for all \(k \ge 1\), and show that the approximation ratio converges to the inverse of the golden ratio \(2/(1+\sqrt{5}) \approx 0.618\) as k goes to infinity. The convergence is extremely fast—the 5-pass Category-Advice algorithm is already within \(0.01\%\) of the inverse of the golden ratio. We then consider two natural greedy algorithms—MinDegree and MinRanking. We analyze MinDegree in the online IID model and show that its approximation ratio is exactly \(1-1/e\). We analyze MinRanking in the priority model and show that this natural algorithm cannot obtain the approximation of the Ranking algorithm in the ROM model.

A. Borodin and D. Pankratov—Research is supported by NSERC.

A. Salehi-Abari—Research was done while the author was a postdoctoral fellow at the University of Toronto. Research was also supported by NSERC.

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Notes

  1. 1.

    The full version of the paper can be found on arXiv [9].

  2. 2.

    In fact, it is easy to see that there is an optimal tie-breaking rule that can be computed by dynamic programming. Unfortunately, the size of the dynamic programming table is exponentially large, and moreover, currently it is not known how to analyze such optimal tie-breaking rules.

  3. 3.

    The Goel and Mehta [19] result is even stronger as it holds for the ROM model.

  4. 4.

    A notable feature of this multi-pass algorithm is that after pass i, the algorithm can deterministically commit to matching a subset of size \(\frac{F_{2i}}{F_{2i+1}}|M|\) where M is a maximum matching. This follows from a certain monotonicity property. See the full version of the paper for details [9].

  5. 5.

    The last property allows us to conclude that the approximation ratio of k-pass Category-Advice converges to the inverse of the golden ratio even when k is allowed to depend on n arbitrarily.

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Authors and Affiliations

  1. University of Toronto, Toronto, ON, Canada

    Allan Borodin & Denis Pankratov

  2. Faculty of Business and IT, UOIT, Oshawa, ON, Canada

    Amirali Salehi-Abari

Authors
  1. Allan Borodin
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  2. Denis Pankratov
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  3. Amirali Salehi-Abari
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Correspondence to Denis Pankratov .

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Editors and Affiliations

  1. The University of Western Ontario, London, Ontario, Canada

    Roberto Solis-Oba

  2. German University of Technology in Oman, Muscat, Oman

    Rudolf Fleischer

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Borodin, A., Pankratov, D., Salehi-Abari, A. (2018). On Conceptually Simple Algorithms for Variants of Online Bipartite Matching. In: Solis-Oba, R., Fleischer, R. (eds) Approximation and Online Algorithms. WAOA 2017. Lecture Notes in Computer Science(), vol 10787. Springer, Cham. https://doi.org/10.1007/978-3-319-89441-6_19

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  • DOI: https://doi.org/10.1007/978-3-319-89441-6_19

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