Computing the \(k\)-resilience of a Synchronized Multi-Robot System

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.