Approximately Supermodular Scheduling Subject to Matroid Constraints

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