A survey on graph problems parameterized above and below guaranteed values

We survey the field of algorithms and complexity for graph problems parameterized above or below guaranteed values. Those problems seek, for a given graph G , a solution whose value is at least g ( G ) + k or at most g ( G ) − k , where g ( G ) is a guarantee on the value that any solution on G takes. The goal is to design algorithms which find such solution in time whose complexity in k is decoupled from that in the guarantee, or to rule out the existence of such algorithms by means of intractability results. We discuss a large number of algorithms and intractability results, and complement them by several open problems.