Hardness of Approximation for Morse Matching

We consider the approximability of maximization and minimization variants of the Morse matching problem, posed as open problems by Joswig and Pfetsch [12]. We establish hardness results for MaxMorse matching and Min-Morse matching. In particular, we show that, for a simplicial complex with n simplices and dimension d ≥ 3, it is NP-hard to approximate Min-Morse matching within a factor of O(n), for any ǫ > 0. Moreover, using an L-reduction from Degree 3 Max-Acyclic Subgraph toMaxMorse matching, we show that it is both NP-hard and UGC-hard to approximate Max-Morse matching for simplicial complexes of dimension d ≥ 2 within certain explicit constant factors.