Optimal Reconstruction Might be Hard

Abstract

Given as input a point set $$\mathcal S $$ that samples a shape $$\mathcal A $$, the condition required for inferring Betti numbers of $$\mathcal A $$ from $$\mathcal S $$ in polynomial time is much weaker than the conditions required by any known polynomial time algorithm for producing a topologically correct approximation of $$\mathcal A $$ from $$\mathcal S $$. Under the former condition which we call the weak precondition, we investigate the question whether a polynomial time algorithm for reconstruction exists. As a first step, we provide an algorithm which outputs an approximation of the shape with the correct Betti numbers under a slightly stronger condition than the weak precondition. Unfortunately, even though our algorithm terminates, its time complexity is unbounded. We then identify at the heart of our algorithm a test which requires answering the following question: given 2 two-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.