Generalized min-up/min-down polytopes

Consider a time horizon and a set of $N$ possible states for a given system. The system must be in exactly one state at a time. In this paper, we generalize classical results on min-up/min-down constraints for a 2-state system to an $N$-state system with $N\ge 3$. The minimum-time constraints enforce that if the system switches to state $i$ at time $t$, then it must remain in state $i$ for a minimum number of time steps. The minimum-time polytope is defined as the convex hull of integer solutions satisfying the minimum-time constraints. A variant of minimum-time constraints is also considered, namely the no-spike constraints. They enforce that if state $i$ is switched on at time $t$, the system must remain on states $j\ge i$ during a minimum time. Symmetrically, they also enforce that if state $i$ is switched off at time $t$, the system must remain on states $j<i$ during a minimum time. The no-spike polytope is defined as the convex hull of integer solutions satisfying the no-spike constraints. For both the minimum-time polytope and the no-spike polytope, we introduce families of valid inequalities. We prove that these inequalities are facet-defining and lead to a complete description of polynomial size for each polytope.