  The question we address in this paper is: which monomial ideals have minimal
cellular resolutions, that is, minimal resolutions obtained from homogenizing
the chain maps of CW-complexes? Velasco gave families of examples of monomial
ideals that do not have minimal cellular resolutions, but those examples have
large minimal generating sets. In this paper, we show that if a monomial ideal
has at most four generators, then the ideal and its (monomial) Artinian
reductions have minimal cellular resolutions. When the ideal is generated by
two monomials, we can give a precise description of the CW-complex supporting
minimal free resolution of the ideal and its Artinian reduction. Also, in this
case, we compute the multigraded Betti numbers, Cohen-Macaulay type and
determine when the corresponding algebra is a level algebra.
