Periodic schedules for linear precedence constraints

We consider the computation of periodic cyclic schedules for linear precedence constraints graph: a linear precedence constraint is defined between two tasks and induces an infinite set of usual precedence constraints between their executions such the the difference of iterations is a linear function.The objective function is the minimization of the maximal period of a task. Firstly, we recall that this problem can be modelled using linear programming. Then, we develop a polynomial algorithm to solve it for unitary graphs, which is a particular class of linear precedence graph.We also show that a periodic schedule may not exists for this special case. In the general case, we compute a decomposition of the graph into unitary components and we suppose that a periodic schedule exists for each of them. We compute lower bounds on the periods and we show that an optimal periodic schedule may not achieve them. Then, we introduce the notion of quasi-periodic schedule, and we prove that this new class of schedule always reach these bounds.