Approximation algorithms for Max Morse Matching

In this paper, we prove that the Max Morse Matching Problem is approximable, thus resolving an open problem posed by Joswig and Pfetsch [1]. For D-dimensional simplicial complexes, we obtain a (D+1)(D2+D+1)-factor approximation ratio using a simple edge reorientation algorithm that removes cycles. For D≥5, we describe a 2D-factor approximation algorithm for simplicial manifolds by processing the simplices in increasing order of dimension. This algorithm leads to 12-factor approximation for 3-manifolds and 49-factor approximation for 4-manifolds. This algorithm may also be applied to non-manifolds resulting in a 1(D+1)-factor approximation ratio. One application of these algorithms is towards efficient homology computation of simplicial complexes. Experiments using a prototype implementation on several datasets indicate that the algorithm computes near optimal results.