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We give a polynomial-time algorithm for OnlineSetCover with a competitiveratio of $O(\log mn)$ when the elements are revealed in random order,essentially matching the best possible offline bound of $O(\log n)$ andcircumventing the $\Omega(\log m \log n)$ lower bound known in adversarialorder. We also extend the result to solving pure covering IPs when constraintsarrive in random order.  The algorithm is a multiplicative-weights-based round-and-solve approach wecall LearnOrCover. We maintain a coarse fractional solution that is neitherfeasible nor monotone increasing, but can nevertheless be rounded online toachieve the claimed guarantee (in the random order model). This gives a newoffline algorithm for SetCover that performs a single pass through theelements, which may be of independent interest.

Random Order Set Cover is as Easy as Offline