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We survey the field of algorithms and complexity for graph problemsparameterized above or below guaranteed values, a research area which waspioneered by Venkatesh Raman. Those problems seek, for a given graph $G$, asolution whose value is at least $g(G)+k$ or at most $g(G)-k$, where $g(G)$ isa guarantee on the value that any solution on $G$ takes. The goal is to designalgorithms which find such solution in time whose complexity in k is decoupledfrom that in the guarantee, or to rule out the existence of such algorithms bymeans of intractability results.  We discuss a large number of algorithms and intractability results, andcomplement them by several open problems.

A Survey on Graph Problems Parameterized Above and Below Guaranteed Values