  We construct cellular resolutions for monomial ideals via discrete Morse
theory. In particular, we develop an algorithm to create homogeneous acyclic
matchings and we call the cellular resolutions induced from these matchings
Barile-Macchia resolutions. These resolutions are minimal for edge ideals of
weighted oriented forests and (most) cycles. As a result, we provide recursive
formulas for graded Betti numbers and projective dimension. Furthermore, we
compare Barile-Macchia resolutions to those created by Batzies and Welker and
some well-known simplicial resolutions. Under certain assumptions, whenever the
above resolutions are minimal, so are Barile-Macchia resolutions.
