  We derive the zero-temperature phase diagram of spin glass models with a
generic fraction of ferromagnetic interactions on the Bethe lattice. We use the
cavity method at the level of one-step replica symmetry breaking (1RSB) and we
find three phases: A replica-symmetric (RS) ferromagnetic one, a magnetized
spin glass one (the so-called mixed phase), and an unmagnetized spin glass one.
We are able to give analytic expressions for the critical point where the RS
phase becomes unstable with respect to 1RSB solutions: we also clarify the
mechanism inducing such a phase transition. Finally we compare our analytical
results with the outcomes of a numerical algorithm especially designed for
finding ground states in an efficient way, stressing weak points in the use of
such numerical tools for discovering RSB effects. Some of the analytical
results are given for generic connectivity.
