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Approximation Schemes for Age of Information Minimization in UAV Grid Patrols

Weiqi Wang, Jin Xu

TL;DR

This work defines edge-level Age of Information (AoI) for UAV grid patrols on general graphs using a Lebesgue-measure formulation and proves a universal lower bound \\overline{\\Delta}(\\mathcal{G}) \\ge \\\frac{1}{2} l(E)^2 that is tight with an Eulerian cycle. With no Eulerian structure, the optimal AoI policy is hard to identify due to an exponential number of periodic routes; two polynomial-time 2-approximation schemes are proposed: a duplicated-edge scheme and a CPP-based scheme, each embedding Eulerian cycles to yield near-optimal patrols. A practical heuristic (heu_cpp) augments these ideas by optimizing edge-visit distribution via a modified Fleury’s algorithm and a potential function, achieving robust performance across random graphs. Numerical results show that evenly distributed edge visits significantly improve AoI relative to distance-focused routes, and that CPP-based strategies perform particularly well in denser graphs, supporting scalable UAV patrol design in hazardous or dynamic environments.

Abstract

Motivated by the critical need for unmanned aerial vehicles (UAVs) to patrol grid systems in hazardous and dynamically changing environments, this study addresses a routing problem aimed at minimizing the time-average Age of Information (AoI) for edges in general graphs. We establish a lower bound for all feasible patrol policies and demonstrate that this bound is tight when the graph contains an Eulerian cycle. For graphs without Eulerian cycles, it becomes challenging to identify the optimal patrol strategy due to the extensive range of feasible options. Our analysis shows that restricting the strategy to periodic sequences still results in an exponentially large number of possible strategies. To address this complexity, we introduce two polynomial-time approximation schemes, each involving a two-step process: constructing multigraphs first and then embedding Eulerian cycles within these multigraphs. We prove that both schemes achieve an approximation ratio of 2. Further, both analytical and numerical results suggest that evenly and sparsely distributing edge visits within a periodic route significantly reduces the average AoI compared to strategies that merely minimize the route travel distance. Building on this insight, we propose a heuristic method that not only maintains the approximation ratio of 2 but also ensures robust performance across varying random graphs.

Approximation Schemes for Age of Information Minimization in UAV Grid Patrols

TL;DR

This work defines edge-level Age of Information (AoI) for UAV grid patrols on general graphs using a Lebesgue-measure formulation and proves a universal lower bound \\overline{\\Delta}(\\mathcal{G}) \\ge \\\frac{1}{2} l(E)^2 that is tight with an Eulerian cycle. With no Eulerian structure, the optimal AoI policy is hard to identify due to an exponential number of periodic routes; two polynomial-time 2-approximation schemes are proposed: a duplicated-edge scheme and a CPP-based scheme, each embedding Eulerian cycles to yield near-optimal patrols. A practical heuristic (heu_cpp) augments these ideas by optimizing edge-visit distribution via a modified Fleury’s algorithm and a potential function, achieving robust performance across random graphs. Numerical results show that evenly distributed edge visits significantly improve AoI relative to distance-focused routes, and that CPP-based strategies perform particularly well in denser graphs, supporting scalable UAV patrol design in hazardous or dynamic environments.

Abstract

Motivated by the critical need for unmanned aerial vehicles (UAVs) to patrol grid systems in hazardous and dynamically changing environments, this study addresses a routing problem aimed at minimizing the time-average Age of Information (AoI) for edges in general graphs. We establish a lower bound for all feasible patrol policies and demonstrate that this bound is tight when the graph contains an Eulerian cycle. For graphs without Eulerian cycles, it becomes challenging to identify the optimal patrol strategy due to the extensive range of feasible options. Our analysis shows that restricting the strategy to periodic sequences still results in an exponentially large number of possible strategies. To address this complexity, we introduce two polynomial-time approximation schemes, each involving a two-step process: constructing multigraphs first and then embedding Eulerian cycles within these multigraphs. We prove that both schemes achieve an approximation ratio of 2. Further, both analytical and numerical results suggest that evenly and sparsely distributing edge visits within a periodic route significantly reduces the average AoI compared to strategies that merely minimize the route travel distance. Building on this insight, we propose a heuristic method that not only maintains the approximation ratio of 2 but also ensures robust performance across varying random graphs.

Paper Structure

This paper contains 16 sections, 10 theorems, 45 equations, 12 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. The AoI-Minimization Problem
  3. Approximation Schemes
  4. Periodic Policy Set $\mathcal{F}_{1}$ and Routes
  5. Lower Bound of AoI in $\mathcal{F}_{1}$
  6. Upper Bound of AoI in $\mathcal{F}_{1}$
  7. The Duplicated Edge Scheme
  8. Approximation Scheme Based on the Chinese Postman Problem (CPP)
  9. A Heuristic Approach and Numerical Studies
  10. Heuristic Approach
  11. Numerical Study
  12. Conclusions and Future Research
  13. Proof of Lemma \ref{['lem:If-an-edge-3']}
  14. Proof of Lemma \ref{['(Upper-bound)-Given']}
  15. Inefficiency of Traversing Fewer Edges
  16. ...and 1 more sections

Key Result

Lemma 1

For any connected graph $\mathcal{G}=(V,E)$, we always have $\overline{\Delta}(\mathcal{G})\geq\frac{1}{2}l(E)^{2}$ with $l(E)=l(\cup E_{i,j})$. The lower bound can be reached when there exists an Eulerian cycle within $\mathcal{G}$.

Figures (12)

  • Figure 1: Demonstrative graph of the grid patrol
  • Figure 2: A snapshot of the AoI on interval $[0,a]$ at $t=\tau+2a$ ($0<\tau<a$).
  • Figure 3: A demonstrative graph for the two visits of $e_0$ if $e_0$ is traversed twice within $R$
  • Figure 4: A snapshot of the age of edge $e_0$ at time $\tau$
  • Figure 5: An example of virtual route $R^{"}$. $R^{"}$ has the same visiting sequence of edges as the actual route $R$. The distance between two visits of $e_0$ is $\frac{l(R)}{2}-l(e_0)$. The virtual route $R^{"}$ is an infeasible route in this case as the second visit of $e_0$ overlaps with the traversal of $e_3$.
  • ...and 7 more figures

Theorems & Definitions (16)

  • Example 1
  • Lemma 1
  • Definition 1
  • Lemma 2
  • Definition 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • ...and 6 more