Online Node- and Edge-Deletion Problems with Advice

Abstract

In online edge- and node-deletion problems the input arrives node by node and an algorithm has to delete nodes or edges in order to keep the input graph in a given graph class Pi at all times. We consider only hereditary properties Pi , for which optimal online algorithms exist and which can be characterized by a set of forbidden subgraphs {{mathcal{F}}} and analyze the advice complexity of getting an optimal solution. We give almost tight bounds on the Delayed Connected{{mathcal{F}}}-Node-Deletion Problem, where all graphs of the family {mathcal{F}} have to be connected and almost tight lower and upper bounds for the DelayedH-Node-Deletion Problem, where there is one forbidden induced subgraph H that may be connected or not. For the DelayedH-Node-Deletion Problem the advice complexity is basically an easy function of the size of the biggest component in H. Additionally, we give tight bounds on the Delayed Connected{mathcal{F}}-Edge-Deletion Problem, where we have an arbitrary number of forbidden connected graphs. For the latter result we present an algorithm that computes the advice complexity directly from {mathcal{F}}. We give a separate analysis for the Delayed ConnectedH-Edge-Deletion Problem, which is less general but admits a bound that is easier to compute.