  Using combinatorics of chains going back to works of Anick, Green, Happel and
Zacharia, we give, for any monomial algebra $A$, an explicit description of its
minimal model. This also provides us with formulas for a canonical
$A_\infty$-structure on the Ext-algebra of the trivial $A$-module. We do this
by exploiting the combinatorics of chains going back to works of Anick, Green,
Happel and Zacharia, and the algebraic discrete Morse theory of J\"ollenbeck,
Welker and Sk\"oldberg. We then show how this result can be used to obtain
models for algebras with a chosen Gr\"obner basis, and briefly outline how to
compute some classical homological invariants with it.
