Reducing off-line to on-line: an example and its applications

We study on-line versions of maximum weighted hereditary subgraph problems for which the instance is revealed in two clusters. We focus on the comparison of these on-line problems with their respective off-line versions. In an earlier paper, we reduced on-line versions to the off-line ones in order to devise competitive analysis for such problems. In this paper, we first devise hardness results pointing out that this previous analysis was tight. Then, we propose a process that allows, for a large class of hereditary problems, to transform an on-line algorithm into an off-line one with improvement of the guarantees. This result can be seen as an inverse version of our previous work. It brings to the fore a hardness gap between on-line and off-line versions of those problems. This result does not apply in the case of maximizing a k-colorable induced subgraph of a given graph. For this problem we point out that, contrary to the first case, the on-line version is almost as well approximated as the off-line one.