@inproceedings{588fa64076314c04b268db0a9fca029b,
title = "Homological reconstruction and simplification in R3",
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.",
keywords = "Homology, NP-hard problems, Persistence",
author = "Dominique Attali and Ulrich Bauer and Olivier Devillers and Marc Glisse and Andr{\'e} Lieutier",
year = "2013",
month = jan,
day = "1",
doi = "10.1145/2493132.2462373",
language = "English",
isbn = "9781450320313",
series = "Proceedings of the Annual Symposium on Computational Geometry",
publisher = "Association for Computing Machinery",
pages = "117--125",
booktitle = "Proceedings of the 29th Annual Symposium on Computational Geometry, SoCG 2013",
note = "29th Annual Symposium on Computational Geometry, SoCG 2013 ; Conference date: 17-06-2013 Through 20-06-2013",
}