  We give an algorithm to determine whether Wilf's conjecture holds for all
numerical semigroups with a given multiplicity $m$, and use it to prove Wilf's
conjecture holds whenever $m \le 18$. Our algorithm utilizes techniques from
polyhedral geometry, and includes a parallelizable algorithm for enumerating
the faces of any polyhedral cone up to orbits of an automorphism group. We also
introduce a new method of verifying Wilf's conjecture via a
combinatorially-flavored game played on the elements of a certain finite poset.
