Determining r-Robustness of Digraphs Using Mixed Integer Linear Programming
For researchers designing resilient multi-agent systems, this provides a more efficient way to determine graph robustness, though it is an incremental improvement over existing methods.
The paper introduces a mixed integer linear programming method to compute the r-robustness of digraphs, which is critical for resilient consensus algorithms. Simulations show it is more efficient than prior algorithms.
Convergence guarantees of many resilient consensus algorithms are 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 high. However, determining the largest integer $r$ for which an arbitrary digraph is $r$-robust is highly nontrivial. This paper introduces a novel method for calculating this value using mixed integer linear programming. The method only requires knowledge of the graph Laplacian matrix, and can be formulated with affine objective and constraints, except for the integer constraint. Integer programming methods such as branch-and-bound can allow both lower and upper bounds on $r$ to be iteratively tightened. Simulations suggest the proposed method demonstrates greater efficiency than prior algorithms.