Optimal Reconstruction Might be Hard

Dominique Attali (1) , André Lieutier (2)

1 GIPSA-AGPIG - GIPSA - Architecture, Géométrie, Perception, Images, Gestes
2 MGMI - Modélisation Géométrique & Multirésolution pour l'Image

Sampling conditions for recovering the homology of a set using topological persistence are much weaker than sampling conditions required by any known polynomial time algorithm for producing a topologically correct reconstruction. Under the former sampling conditions which we call {\em weak sampling conditions}, we give an algorithm that outputs a topologically correct reconstruction. Unfortunately, even though the algorithm terminates, its time complexity is unbounded. Motivated by the question of knowing if a polynomial time algorithm for reconstruction exists under the weak sampling conditions, we identify at the heart of our algorithm a test which requires answering the following question: given two 2-dimensional simplicial complexes $L \subset K$, does there exist a simplicial complex containing $L$ and contained in $K$ which realizes the persistent homology of $L$ into $K$? We call this problem the homological simplification of the pair $(K,L)$ and prove that this problem is NP-complete, using a reduction from 3SAT.

Mots clés: Shape reconstruction; topological persistence; homological simplification; sampling conditions; 3SAT; NP-completeness

Domaines: Géométrie algorithmique [cs.CG]

Dominique Attali, André Lieutier. Optimal Reconstruction Might be Hard. SoCG 2010 - 26th Annual Symposium on Computational Geometry, Jun 2010, Snowbird, Utah, United States. pp.334-343, ⟨10.1145/1810959.1811014⟩. ⟨hal-00468602⟩