Determining r-Robustness of Digraphs Using Mixed Integer Linear Programming

TL;DR

This paper presents a new mixed integer linear programming approach to efficiently determine the maximum r-robustness of directed graphs, which is crucial for resilient consensus algorithms.

Contribution

It introduces a novel MILP-based method that computes r-robustness using only the graph Laplacian, improving efficiency over previous algorithms.

Findings

The method accurately computes r-robustness bounds.

Simulations show improved efficiency over prior algorithms.

The approach is applicable to arbitrary digraphs using only Laplacian data.

Abstract

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.