  In this paper, we prove that the Max-Morse Matching Problem is approximable,
thus resolving an open problem posed by Joswig and Pfetsch. We describe two
different approximation algorithms for the Max-Morse Matching Problem. For
$D$-dimensional simplicial complexes, we obtain a
$\frac{(D+1)}{(D^2+D+1)}$-factor approximation ratio using a simple edge
reorientation algorithm that removes cycles. Our second result is an algorithm
that provides a $\frac{2}{D}$-factor approximation for simplicial manifolds by
processing the simplices in increasing order of dimension. 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.
