  There has been an increase in the use of resilient control algorithms based
on the graph theoretic properties of $r$- and $(r,s)$-robustness. These
algorithms guarantee consensus of normally behaving agents in the presence of a
bounded number of arbitrarily misbehaving agents if the values of the integers
$r$ and $s$ are sufficiently large. However, determining an arbitrary graph's
robustness is a highly nontrivial problem. This paper introduces a novel method
for determining the $r$- and $(r,s)$-robustness of digraphs using mixed integer
linear programming; to the best of the authors' knowledge it is the first time
that mixed integer programming methods have been applied to the robustness
determination problem. The approach only requires knowledge of the graph
Laplacian matrix, and can be formulated with binary integer variables. Mixed
integer programming algorithms such as branch-and-bound are used to iteratively
tighten the lower and upper bounds on $r$ and $s$. Simulations are presented
which compare the performance of this approach to prior robustness
determination algorithms.
