On the Advice Complexity of Online Edge- and Node-Deletion Problems

Abstract

We consider a model of online graph modification problems where the input graph is read piecewise in an adversarial order and the algorithm has to modify the graph by deleting vertices or edges in order to keep it inside some fixed graph class \(\varPi \). These deletions cannot be taken back later. We analyze the least number of advice bits that enable an online algorithm to perform the same number of deletions as an optimal offline algorithm. We consider only hereditary properties \(\varPi \), for which optimal online algorithms exist and which can be characterized by a set of forbidden subgraphs \(\mathcal {F}\). It is clear that, if \(\mathcal {F}\) is finite, then the number of required advice bits is at most linear in the size of an optimal solution. We prove lower and upper bounds based on the structure of the graphs in \(\mathcal {F}\), often determining the complexity exactly or nearly exactly. The techniques also work for infinite \(\mathcal {F}\) and we can determine whether then the advice complexity can be bounded by a function in the optimal solution or not. For node-deletion problems we characterize the advice complexity exactly for all cases and for edge-deletion problems at least for the case of a single forbidden induced subgraph.