Multi-Agent Search-Type Problems on Polygons

TL;DR

The paper tackles multi-agent search with discrete target locations arranged as regular polygons, introducing the Polygon Priority Evacuation problem $\sc{PE}^{n}_{k}$ and related disk variants under wireless communication. It combines nonlinear program (NLP) formulations with linear program (LP) relaxations to derive upper and lower bounds for $\sc{PE}^{n}_{k}$, $\sc{DE}_{k}$, and $\sc{PE}^{n}(w)$, $\sc{DE}(w)$ across a range of $n$ and $k$ (notably up to $n\le 13$ and $k\le 4$). The authors achieve improved lower bounds for $\sc{DE}_{1}$ and $\sc{DE}_{2}$, and, through new analyses of $11$- and $12$-gon cases, provide tighter or near-tight bounds for the $w$-weighted disk and polygon search problems, reducing the gap between upper and lower bounds significantly. The results offer near-optimal search trajectories and highlight the strength of LP relaxations in yielding tight bounds for complex continuous-discrete hybrid search problems, while also pointing to open questions about embeddability gaps and extensions to other communication models.

Abstract

We present several advancements in search-type problems for fleets of mobile agents operating in two dimensions under the wireless model. Potential hidden target locations are equidistant from a central point, forming either a disk (infinite possible locations) or regular polygons (finite possible locations). Building on the foundational disk evacuation problem, the disk priority evacuation problem with $k$ Servants, and the disk $w$-weighted search problem, we make improvements on several fronts. First we establish new upper and lower bounds for the $n$-gon priority evacuation problem with $1$ Servant for $n \leq 13$, and for $n_k$-gons with $k=2, 3, 4$ Servants, where $n_2 \leq 11$, $n_3 \leq 9$, and $n_4 \leq 10$, offering tight or nearly tight bounds. The only previous results known were a tight upper bound for $k=1$ and $n=6$ and lower bounds for $k=1$ and $n \leq 9$. Second, our work improves the best lower bound known for the disk priority evacuation problem with $k=1$ Servant from $4.46798$ to $4.64666$ and for $k=2$ Servants from $3.6307$ to $3.65332$. Third, we improve the best lower bounds known for the disk $w$-weighted group search problem, significantly reducing the gap between the best upper and lower bounds for $w$ values where the gap was largest. These improvements are based on nearly tight upper and lower bounds for the $11$-gon and $12$-gon $w$-weighted evacuation problems, while previous analyses were limited only to lower bounds and only to $7$-gons.

Figures (35)

• Figure 1: Known upper bounds compared to our improved lower bounds for $\sc{DE}({w})$. The blue curve (labeled $a$) depicts the best upper bound known GW24-iwoca. The green curve (labeled $c$) depicts the previously best lower bound known, by a reduction to $\sc{PE}^{7}({w})$. The orange line (labeled $b$) depicts a universal lower bound of $1+\pi$ also proved in GW24-iwoca. The purple and black curves (labeled $d,e$, respectively) are new lower bounds on $\sc{DE}({w})$ by a reductions to $\sc{PE}^{11}({w})$ and $\sc{PE}^{12}({w})$, respectively (hence the maximum of them applies). The lower bounds were calculated for values of $w\in[0,1]$ starting from $0$ and with step size $0.01$.
• Figure 2: Upper and lower bound obtained for $\sc{PE}^{11}({w})$ (left hand-side) and $\sc{PE}^{12}({w})$ (right hand-side). Starting from $w=0$, the lower bounds were obtained with $w$ step size of $0.01$ and the upper bounds with $w$ step size of $0.02$.
• Figure 3: A search trajectory for $\sc{PE}^{9}_{1}$. Trajectory details can be found in Table \ref{['9gonk1tab']}. The Queen waits at $V_3$ for an amount of time such that the Queen arrives at $Q^{(5)}$ at time $2e_9$. The position of $Q^{(5)}$ has been adjusted for clarity.
• Figure 4: A search trajectory for $\sc{PE}^{3}_{1}$. Trajectory details can be found in Table \ref{['3gonk1tab']}.
• Figure 5: A search trajectory for $\sc{PE}^{4}_{1}$. Trajectory details can be found in Table \ref{['4gonk1tab']}.

Theorems & Definitions (13)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 3.2: Introduced in GW24-iwoca, for $k=1$
• proof
• Lemma 3.3
• proof
• Corollary 3.4
• Example 5.1: Upper Bound to $\sc{PE}^{9}_{1}$