Approximately Supermodular Scheduling Subject to Matroid Constraints

Control scheduling refers to the problem of assigning agents or actuators to act upon a dynamical system at specific times so as to minimize a quadratic control cost, such as the objectives of the linear-quadratic-Gaussian (LQG) or linear quadratic regulator problems. When budget or operational constraints are imposed on the schedule, this problem is in general NP-hard and its solution can therefore only be approximated even for moderately sized systems. The quality of this approximation depends on the structure of both the constraints and the objective. This article shows that greedy control scheduling is near-optimal when the constraints can be written as an intersection of matroids, algebraic structures that encode requirements such as limits on the number of agents deployed per time slot, total number of actuator uses, and duty cycle restrictions. To do so, it proves that the LQG cost function is
$\alpha$
-supermodular and provides new
$\alpha /(\alpha + P)$
-optimality certificates for the greedy minimization of such functions over an intersection of
$P$
matroids. These certificates are shown to approach the
$1/(1+P)$
guarantee of supermodular functions in relevant settings. These results support the use of greedy algorithms in nonsupermodular quadratic control problems as opposed to typical heuristics such as convex relaxations and surrogate figures of merit, e.g., the
$\log \det$
of the controllability Gramian.