  We study methods for finding the solution set of a generic system in a family
of polynomial systems with parametric coefficients. We present a framework for
describing monodromy based solvers in terms of decorated graphs. Under the
theoretical assumption that monodromy actions are generated uniformly, we show
that the expected number of homotopy paths tracked by an algorithm following
this framework is linear in the number of solutions. We demonstrate that our
software implementation is competitive with the existing state-of-the-art
methods implemented in other software packages.
