We give a polynomial-time algorithm for OnlineSetCover with a competitive ratio of $O(\log mn)$ when the elements are revealed in random order, essentially matching the best possible offline bound of $O(\log n)$ and circumventing the $\Omega(\log m \log n)$ lower bound known in adversarial order. We also extend the result to solving pure covering IPs when constraints arrive in random order. The algorithm is a multiplicative-weights-based round-and-solve approach we call LearnOrCover. We maintain a coarse fractional solution that is neither feasible nor monotone increasing, but can nevertheless be rounded online to achieve the claimed guarantee (in the random order model). This gives a new offline algorithm for SetCover that performs a single pass through the elements, which may be of independent interest.

翻译：当元素按随机顺序披露时,我们给在线SetCover提供一种多边时间算法,其竞争性比率为$O(\log mnn),基本上匹配最佳的离线约束$O(\log n),绕过在对抗性排序中已知的较低约束$Omega(\log m\log n) 。我们还将结果扩展至在限制到达随机顺序时解决纯覆盖IP。这种算法是一种基于多倍增重量的圆和溶解方法,我们称之为LearOrCover。我们维持一种粗略的分数解决方案,既不可行,也不增加单质,但可以四舍五入在线实现(随机顺序模式)的所谓担保。这为SetCover提供了一种新的离线算法,通过元素进行单次传输,这些元素可能具有独立的兴趣。