# Determining r- and (r,s)-Robustness of Digraphs Using Mixed Integer Linear Programming

## TL;DR

This paper presents a novel mixed integer linear programming method to efficiently determine the $r$- and $(r,s)$-robustness of directed graphs, a key property for resilient control algorithms, using only the graph Laplacian.

## Contribution

It introduces the first application of mixed integer programming to compute graph robustness, providing an efficient and scalable approach based on the Laplacian matrix.

## Key findings

- The method accurately determines robustness parameters in simulations.
- It outperforms prior algorithms in computational efficiency.
- The approach is applicable to various directed graphs.

## Abstract

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.