The Online Set Cover Problem

Let $X=\{1,2,\ldots,n\}$ be a ground set of $n$ elements, and let ${\cal S}$ be a family of subsets of $X$, $|{\cal S}|=m$, with a positive cost $c_S$ associated with each $S\in{\cal S}$. Consider the following online version of the set cover problem, described as a game between an algorithm and an adversary. An adversary gives elements to the algorithm from $X$ one by one. Once a new element is given, the algorithm has to cover it by some set of ${\cal S}$ containing it. We assume that the elements of $X$ and the members of ${\cal S}$ are known in advance to the algorithm; however, the set $X'\subseteq X$ of elements given by the adversary is not known in advance to the algorithm. (In general, $X'$ may be a strict subset of $X$.) The objective is to minimize the total cost of the sets chosen by the algorithm. Let ${\cal C}$ denote the family of sets in ${\cal S}$ that the algorithm chooses. At the end of the game the adversary also produces (offline) a family of sets ${\cal C}_{OPT}$ that covers $X'$. The performance of the algorithm is the ratio between the cost of ${\cal C}$ and the cost of ${\cal C}_{OPT}$. The maximum ratio, taken over all input sequences, is the competitive ratio of the algorithm. We present an $O(\log m\log n)$ competitive deterministic algorithm for the problem and establish a nearly matching $\Omega\bigl(\frac{\log n\log m}{\log\log m+\log\log n}\bigr)$ lower bound for all interesting values of $m$ and $n$. The techniques used are motivated by similar techniques developed in computational learning theory for online prediction (e.g., the WINNOW algorithm) together with a novel way of converting a fractional solution into a deterministic online algorithm.