Computing the $k$-resilience of a Synchronized Multi-Robot System

TL;DR

This paper introduces the concept of $k$-resilience in synchronized multi-robot systems, analyzing how many robot failures can cause starvation among remaining robots, and provides algorithms to compute this resilience.

Contribution

It defines the $k$-resilience metric for synchronized multi-robot systems and presents algorithms for its computation, including complexity results and efficient methods for tree communication graphs.

Findings

Computing $k$-resilience is NP-hard when $k$ varies with input.

Algorithms are provided for constant $k$ values in general graphs.

More efficient algorithms are developed for tree-structured communication graphs.

Abstract

We study an optimization problem that arises in the design of covering strategies for multi-robot systems. Consider a team of $n$ cooperating robots traveling along predetermined closed and disjoint trajectories. Each robot needs to periodically communicate information to nearby robots. At places where two trajectories are within range of each other, a communication link is established, allowing two robots to exchange information, provided they are "synchronized", i.e., they visit the link at the same time. In this setting a communication graph is defined and a system of robots is called \emph{synchronized} if every pair of neighbors is synchronized. If one or more robots leave the system, then some trajectories are left unattended. To handle such cases in a synchronized system, when a live robot arrives to a communication link and detects the absence of the neighbor, it shifts to the neighboring trajectory to assume the unattended task. If enough robots leave, it may occur that a live robot enters a state of \emph{starvation}, failing to permanently meet other robots during flight. To measure the tolerance of the system under this phenomenon we define the \emph{$k$-resilience} as the minimum number of robots whose removal may cause $k$ surviving robots to enter a state of starvation. We show that the problem of computing the $k$-resilience is NP-hard if $k$ is part of the input, even if the communication graph is a tree. We propose algorithms to compute the $k$-resilience for constant values of $k$ in general communication graphs and show more efficient algorithms for systems whose communication graph is a tree.