AI 中文总结
该研究针对顶点到达下的在线顶点覆盖问题，将Wang和Wong的构造扩展后证明了新的竞争比下界约为1.824，改进了此前的约1.753下界，还验证了该下界是递推式的精确极限。
AI 中文摘要
我们证明，对于一般顶点到达下的在线顶点覆盖问题，即使在二分图上且面对无知觉对手，任何随机的整算法或分数算法都无法达到严格低于 $1+\sqrt{e}/2\approx1.824360635$ 的竞争比，这改进了此前约为1.753的下界。我们的证明将Wang和Wong的完全二分图交替构造扩展到任意次交替，所得对手由一个单调整递推式描述。若该递推式从未违反竞争预算，其迭代会收敛到一个可积不动点；对所有此类不动点进行分类，可推导出超额比至少为 $\sqrt{e}/2$。截断的离散递推式与黎曼和论证将每一个严格的连续违反转化为有限的、依赖算法但与实现无关的输入。我们还展示了一个关键不动点，表明 $1+\sqrt{e}/2$ 是该齐次完全二分图递推式的精确极限，而非数值假象。
英文摘要
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.
