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：我们提出了一种多项式时间算法，用于解决随机顺序元素出现时的在线集合覆盖问题，其竞争比为$O(\log mn)$，这本质上匹配了离线情形下的最优界$O(\log n)$，并规避了已知在对抗顺序下存在的$\Omega(\log m \log n)$下界。我们还将该结果推广到约束条件以随机顺序到达时的纯覆盖整数规划问题。该算法基于乘性权重的轮次求解框架，我们称之为LearnOrCover。我们维护一个既不可行也不单调递增的粗略分数解，但该解仍可在在线环境下通过舍入操作实现所宣称的保证（在随机顺序模型中）。这为集合覆盖问题提供了一种仅需单遍遍历元素的新离线算法，该算法可能具有独立的研究价值。