Hardness of Approximation for Morse Matching

Discrete Morse theory has emerged as a powerful tool for a wide range of problems, including the computation of (persistent) homology. In this context, discrete Morse theory is used to reduce the problem of computing a topological invariant of an input simplicial complex to computing the same topological invariant of a (significantly smaller) collapsed cell or chain complex. Consequently, devising methods for obtaining gradient vector fields on complexes to reduce the size of the problem instance has become an emerging theme over the last decade. While computing the optimal gradient vector field on a simplicial complex is NP-hard, several heuristics have been observed to compute near-optimal gradient vector fields on a wide variety of datasets. Understanding the theoretical limits of these strategies is therefore a fundamental problem in computational topology. In this paper, we consider the approximability of maximization and minimization variants of the Morse matching problem, posed as open problems by Joswig and Pfetsch. We establish hardness results for Max-Morse matching and Min-Morse matching. In particular, we show that, for a simplicial complex with n simplices and dimension $d \leq 3$, it is NP-hard to approximate Min-Morse matching within a factor of $O(n^{1-\epsilon})$, for any $\epsilon > 0$. Moreover, using an L-reduction from Degree 3 Max-Acyclic Subgraph to Max-Morse matching, we show that it is both NP-hard and UGC-hard to approximate Max-Morse matching for simplicial complexes of dimension $d \leq 2$ within certain explicit constant factors.