Optimality of non-conservative driving for finite-time processes with discrete states

An optimal finite-time process drives a given initial distribution to a given final one in a given time at the lowest cost as quantified by total entropy production. We prove that for system with discrete states this optimal process involves non-conservative driving, i.e., a genuine driving affinity, in contrast to the case of system with continuous states. In a multicyclic network, the optimal driving affinity is bounded by the number of states within each cycle. If the driving affects forward and backwards rates non-symmetrically, the bound additionally depends on a structural parameter characterizing this asymmetry.