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arXiv · 2608.09210

A New Lower Bound for Online Vertex Cover under Vertex Arrivals

Tianhang Lu
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Abstract

We prove that no randomized integral or fractional algorithm for online vertex cover under general vertex arrivals achieves a competitive ratio strictly below $1+\sqrt{e}/2\approx1.824360635$, even on bipartite graphs and against an oblivious adversary. This improves the previous lower bound of approximately $1.753$. Our proof extends the complete-bipartite alternating construction of Wang and Wong to an arbitrary number of alternations. The resulting adversary is described by a monotone integral recurrence. If the recurrence never violates the competitive budget, its iterates converge to an integrable fixed point; classifying all such fixed points forces the excess ratio to be at least $\sqrt{e}/2$. A truncated discrete recurrence and a Riemann-sum argument convert every strict continuous violation into a finite, algorithm-dependent but realization-oblivious input. We also exhibit a critical fixed point showing that $1+\sqrt{e}/2$ is the exact limit of this homogeneous complete-bipartite recurrence, rather than a numerical artifact.

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Tianhang Lu. 2026-08-10. A New Lower Bound for Online Vertex Cover under Vertex Arrivals. https://arxiv.org/abs/2608.09210

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