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

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.

💡 Research Summary

The paper addresses a fundamental problem in persistent robotic monitoring: given a known set of locations that must be observed repeatedly, how should a robot traverse the environment so that the most important location experiences the smallest worst‑case delay between successive observations? The environment is abstracted as a weighted undirected graph G = (V,E). Each vertex v ∈ V represents a region of interest, a non‑negative weight w(v) quantifies its importance, and each edge e ∈ E carries a length ℓ(e) that corresponds to travel time. The robot repeatedly executes a closed walk W on G. For any vertex v, the latency L(v) is the maximum time elapsed between two consecutive visits to v while traversing W. The weighted latency is defined as w(v)·L(v). The objective, called the Weighted Latency Minimization (WLM) problem, is to find a closed walk that minimizes the maximum weighted latency over all vertices, i.e., min_W max_{v∈V} w(v)·L(v).

The authors first prove that WLM is NP‑hard. By reducing from the classic Set‑Cover problem they show that when vertex weights vary, deciding whether a walk exists with weighted latency ≤ K is as hard as covering a universe with a limited number of sets. Consequently, no polynomial‑time algorithm can guarantee an optimal solution unless P = NP. This hardness result also implies that standard TSP or min‑max latency formulations, which assume uniform importance, are insufficient for many real‑world monitoring tasks where some sites (e.g., high‑crime neighborhoods) must be revisited more frequently.

Given the intractability, the paper proposes two logarithmic‑factor approximation algorithms.

1. O(log n)‑approximation – The algorithm partitions vertices into O(log n) weight classes using a geometric progression (e.g., weights in