Competitive Perimeter Defense in Tree Environments

TL;DR

This work studies online perimeter defense in rooted full tree environments where intruders enter from leaves and move toward a perimeter defined by distance $\rho$ from the root, while a single defender with unit speed tries to intercept them. Using competitive analysis, it derives fundamental limits that bound any online policy relative to the offline optimum, revealing regimes where finite, 2-, or 3/2-competitive performance is impossible or achievable. It then proposes three online algorithms—Sweeping, Stay At Perimeter (SaP), and Compare and Subtree Sweep (CaSS)—each with provable competitiveness under distinct parameter regimes, and provides numerical visualizations of the resulting trade-offs. The results illuminate a spectrum of strategies balancing full-tree coverage versus targeted subtree sweeps, highlighting how intruder velocity and tree parameters shape algorithmic efficacy. The findings guide practical deployment of a single defender in tree-like environments and point to future work on non-full trees and multi-defender extensions with tighter benchmarks.

Abstract

We consider a perimeter defense problem in a rooted full tree graph environment in which a single defending vehicle seeks to defend a set of specified vertices, termed as the perimeter from mobile intruders that enter the environment through the tree's leaves. We adopt the technique of competitive analysis to characterize the performance of an online algorithm for the defending vehicle. We first derive fundamental limits on the performance of any online algorithm relative to that of an optimal offline algorithm. Specifically, we give three fundamental conditions for finite, 2, and 3/2 competitive ratios in terms of the environment parameters. We then design and analyze three classes of online algorithms that have provably finite competitiveness under varying environmental parameter regimes. Finally, we give a numerical visualization of these regimes to better show the comparative strengths and weaknesses of each algorithm.

Figures (5)

• Figure 1: An environment with depth $d=2$, perimeter depth $\rho=1$, and branching factor $\delta=2$. The small purple dots represent the intruders trying to reach perimeter vertices $p_1$ and $p_2$ after arriving from intruder entrances $\ell_1$ and $\ell_4$. The defending vehicle is represented by the blue triangle.
• Figure 2: Breakdown of an environment as perceived by the Stay at Perimeter (SaP) algorithm. Here SaP considers the two branches rooted at $p_1$ and $p_2$. The regions of the subtrees considered in each epoch are highlighted. As long as the defender is located at $p_1$ all intruders from the left subtree will be captured.
• Figure 3: The Compare and Subtree Sweep algorithm will follow the path in purple when $s=1$ and $|M_{1}| > |M_{2}|$ (shown as the shaded regions). As $\rho=1$ in this environment, there is only a single valid value for $s$.
• Figure 4: Parameter regimes for fundamental limits, Sweeping algorithm, and Stay at Perimeter Algorithm for varying values of $\rho$ on a full tree with depth $d=20$ and branching factor $\delta=3$. The SaP algorithm is said to be $\mathcal{O}(\delta^{\rho})$ competitive as its competitiveness is given by $\frac{3 \cdot \delta^{\rho} -1}{2}$, where the $\mathcal{O}(\cdot)$ refers to the Landau notation.
• Figure 5: Parameter regimes for Compare and Subtree Sweep Algorithm and Sweeping Algorithm for all values of $\rho$ on a full tree with depth $d=5$ and branching factor $\delta=2$.

Theorems & Definitions (19)

• Definition 1: Online Algorithm
• Definition 2: Offline Algorithm
• Definition 3: Competitive Ratio, bajaj2021competitivebajaj2022competitive
• Definition 4: Descendant Vertices
• Definition 5: Branch of a Tree Graph
• Definition 6: Branch Entrances
• Theorem III.1
• Theorem III.2
• proof
• Theorem III.3