We consider the problem of deciding whether the persistent homology group of a simplicial pair can be realized as the homology of some complex X with . We show that this problem is NP-complete even if K is embedded in .
As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard.