Persistent Monitoring in Discrete Environments: Minimizing the Maximum Weighted Latency Between Observations

TL;DR

This paper addresses the challenge of planning robot patrol paths in known environments to minimize the maximum weighted latency between visits, introducing approximation algorithms for an NP-hard problem.

Contribution

It formulates the problem of minimizing maximum weighted latency in robot monitoring, proves its computational hardness, and proposes two approximation algorithms with proven bounds.

Findings

Algorithms scale to thousands of vertices in simulations.

Proposed methods effectively reduce maximum weighted latency.

Case study demonstrates practical application in city crime patrolling.

Abstract

In this paper, we consider the problem of planning a path for a robot to monitor a known set of features of interest in an environment. We represent the environment as a graph with vertex weights and edge lengths. The vertices represent regions of interest, edge lengths give travel times between regions, and the vertex weights give the importance of each region. As the robot repeatedly performs a closed walk on the graph, we define the weighted latency of a vertex to be the maximum time between visits to that vertex, weighted by the importance (vertex weight) of that vertex. Our goal is to find a closed walk that minimizes the maximum weighted latency of any vertex. We show that there does not exist a polynomial time algorithm for the problem. We then provide two approximation algorithms; an $O(\log n)$-approximation algorithm and an $O(\log \rho_G)$-approximation algorithm, where $\rho_G$ is the ratio between the maximum and minimum vertex weights. We provide simulation results which demonstrate that our algorithms can be applied to problems consisting of thousands of vertices, and a case study for patrolling a city for crime.