@inproceedings{2812,
  abstract     = {We consider the problem of deciding whether the persistent homology group of a simplicial pair (K, L) can be realized as the homology H* (X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in ℝ3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on S3 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.},
  author       = {Attali, Dominique and Bauer, Ulrich and Devillers, Olivier and Glisse, Marc and Lieutier, André},
  booktitle    = {Proceedings of the 29th annual symposium on Computational Geometry},
  location     = {Rio de Janeiro, Brazil},
  pages        = {117 -- 125},
  publisher    = {ACM},
  title        = {{Homological reconstruction and simplification in R3}},
  doi          = {10.1145/2462356.2462373},
  year         = {2013},
}