Random Order Online Set Cover is as Easy as Offline

Anupam Gupta, Gregory Kehne, Roie Levin

Abstract: We give a polynomial-time algorithm for Online-SetCover with a competitive ratio of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log mn)$</tex> when the elements are revealed in random order, matching the best possible offline bound of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log n)$</tex> when the number of sets <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m$</tex> is polynomial in the number of elements <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> , and circumventing the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Omega(\log m \log n)$</tex> 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.