The ALT auction algorithm for bipartite matching works without freezing, admits an augmenting-path analysis, and requires Θ(1/ε²) iterations even on paths.
Semi-streaming bipartite matching in fewer passes and optimal space
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On the Assadi Liu Tarjan Auction Algorithm for Bipartite Matching: Simplification, Alternative Analysis, and Hard Instance
The ALT auction algorithm for bipartite matching works without freezing, admits an augmenting-path analysis, and requires Θ(1/ε²) iterations even on paths.