Random Order Online Set Cover is as Easy as Offline

We give a polynomial-time algorithm for Online-SetCover with a competitive ratio of
$O(\log mn)$
when the elements are revealed in random order, matching the best possible offline bound of
$O(\log n)$
when the number of sets
$m$
is polynomial in the number of elements
$n$
, 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.