  Establishing whether an algebra is quasi-hereditary or not is, in general, a
difficult problem. In this paper we introduce a sufficient criterion to
determine whether a general finite dimensional algebra is quasi-hereditary by
showing that the question can be reduced to showing that a closely associated
monomial algebra is quasi-hereditary. For monomial algebras, we give an
explicit, easily verifiable, necessary and sufficient criterion to determine
whether it is quasi-hereditary.
