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 in...
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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 in this model is 1/2, which is achieved by any greedy algorithm. Durr et al. (2016) presented a 2-pass algorithm Category-Advice with 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 >= 1, and show that the approximation ratio converges quickly to the inverse of the golden ratio 2/(1+5)approximate to 0.618\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2/(1+\sqrt {5}) \approx 0.618$\end{document} as k goes to infinity. We then consider a natural adaptation of a well-known offline MinGreedy algorithm to the online stochastic IID model, which we call MinDegree. In spite of excellent empirical performance of MinGreedy, it was recently shown to have approximation ratio 1/2 in the adversarial offline setting - the approximation ratio achieved by any greedy algorithm. Our result in the online known IID model is, in spirit, similar to the offline result, but the proof is different. We show that MinDegree cannot achieve an approximation ratio better than 1 - 1/e, which is guaranteed by any consistent greedy algorithm in the known IID model. Finally, following the work in Besser and Poloczek (Algorithmica 2017(1), 201-234, 2017), we depart from an adversarial or stochastic ordering and investigate a natural randomized algorithm (MinRanking) in the priority model. Although the priority model allows the algorithm to choose the input ordering in a general but well defined way, this natural algorithm cannot obtain the approximation of the Ranking algorithm in the ROM model.
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