For both the edge deletion heuristic and the
maximum-degree greedy heuristic, we study
the problem of recognizing those graphs for which
that heuristic can approximate
the size of a minimum vertex cover within a constant factor
of r, where r is a fixed rational number.
Our main results are
that these problems are complete for the class of problems solvable via
parallel access to NP.
To achieve these main results, we also show that
the restriction of the vertex cover problem to those graphs for which either
of these heuristics can find an optimal solution remains NP-hard.