Online Node- and Edge-Deletion Problems with Advice

摘要

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 F and analyze the advice complexity of getting an optimal solution. We give almost tight bounds on the DELAYED CONNECTED F-NODE-DELETION PROBLEM, where all graphs of the family F have to be connected and almost tight lower and upper bounds for the DELAYED H-NODE-DELETION PROBLEM, where there is one forbidden induced subgraph H that may be connected or not. For the DELAYED H-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 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 F. We give a separate analysis for the Delayed Connected H-Edge-Deletion Problem, which is less general but admits a bound that is easier to compute.