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Computer Science > Data Structures and Algorithms

arXiv:2511.16025 (cs)
[Submitted on 20 Nov 2025]

Title:Optimal Online Bipartite Matching in Degree-2 Graphs

Authors:Amey Bhangale, Arghya Chakraborty, Prahladh Harsha
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Abstract:Online bipartite matching is a classical problem in online algorithms and we know that both the deterministic fractional and randomized integral online matchings achieve the same competitive ratio of $1-\frac{1}{e}$. In this work, we study classes of graphs where the online degree is restricted to $2$. As expected, one can achieve a competitive ratio of better than $1-\frac{1}{e}$ in both the deterministic fractional and randomized integral cases, but surprisingly, these ratios are not the same. It was already known that for fractional matching, a $0.75$ competitive ratio algorithm is optimal. We show that the folklore \textsc{Half-Half} algorithm achieves a competitive ratio of $\eta \approx 0.717772\dots$ and more surprisingly, show that this is optimal by giving a matching lower-bound. This yields a separation between the two problems: deterministic fractional and randomized integral, showing that it is impossible to obtain a perfect rounding scheme.
Subjects: Data Structures and Algorithms (cs.DS)
MSC classes: 68W20, 68R10, 90C27
Cite as: arXiv:2511.16025 [cs.DS]
  (or arXiv:2511.16025v1 [cs.DS] for this version)
  https://doi.org/10.48550/arXiv.2511.16025

Submission history

From: Arghya Chakraborty [view email]
[v1] Thu, 20 Nov 2025 04:12:49 UTC (147 KB)
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