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Weighted Group Search on the Disk & Improved Lower Bounds for Priority Evacuation

Konstantinos Georgiou, Xin Wang

TL;DR

This work generalizes two-agent search on the unit disk by introducing weighted group search with cost $g_w(x,y)=wx+y$ and $w\in[0,1]$, unifying evacuation, priority evacuation, and weighted objectives. It develops a novel LP-relaxation framework, inspired by metric embedding, to derive lower bounds across all $w$ and applies it to improve the priority-evacuation lower bound to $4.56798$, substantially narrowing the upper-lower bound gap. The authors also construct $w$-dependent upper bounds via generalized Detour-like trajectories, establishing explicit formulas such as $1+ d_w + 2\sin(d_w)/(w+1)$ with $d_w$ defined piecewise by $\alpha_w$, $\beta_w$, and a critical $w_0\approx 0.0456911$. Overall, the paper provides a cohesive methodology for tight lower bounds and practical algorithms across the spectrum of weighted objectives, advancing the theoretical understanding and enabling sharper performance guarantees for disk-based search and evacuation problems.

Abstract

We consider \emph{weighted group search on a disk}, which is a search-type problem involving 2 mobile agents with unit-speed. The two agents start collocated and their goal is to reach a (hidden) target at an unknown location and a known distance of exactly 1 (i.e., the search domain is the unit disk). The agents operate in the so-called \emph{wireless} model that allows them instantaneous knowledge of each others findings. The termination cost of agents' trajectories is the worst-case \emph{arithmetic weighted average}, which we quantify by parameter $w$, of the times it takes each agent to reach the target, hence the name of the problem. Our work follows a long line of research in search and evacuation, but quite importantly it is a variation and extension of two well-studied problems, respectively. The known variant is the one in which the search domain is the line, and for which an optimal solution is known. Our problem is also the extension of the so-called \emph{priority evacuation}, which we obtain by setting the weight parameter $w$ to $0$. For the latter problem the best upper/lower bound gap known is significant. Our contributions for weighted group search on a disk are threefold. \textit{First}, we derive upper bounds for the entire spectrum of weighted averages $w$. Our algorithms are obtained as a adaptations of known techniques, however the analysis is much more technical. \textit{Second}, our main contribution is the derivation of lower bounds for all weighted averages. This follows from a \emph{novel framework} for proving lower bounds for combinatorial search problems based on linear programming and inspired by metric embedding relaxations. \textit{Third}, we apply our framework to the priority evacuation problem, improving the previously best lower bound known from $4.38962$ to $4.56798$, thus reducing the upper/lower bound gap from $0.42892$ to $0.25056$.

Weighted Group Search on the Disk & Improved Lower Bounds for Priority Evacuation

TL;DR

This work generalizes two-agent search on the unit disk by introducing weighted group search with cost and , unifying evacuation, priority evacuation, and weighted objectives. It develops a novel LP-relaxation framework, inspired by metric embedding, to derive lower bounds across all and applies it to improve the priority-evacuation lower bound to , substantially narrowing the upper-lower bound gap. The authors also construct -dependent upper bounds via generalized Detour-like trajectories, establishing explicit formulas such as with defined piecewise by , , and a critical . Overall, the paper provides a cohesive methodology for tight lower bounds and practical algorithms across the spectrum of weighted objectives, advancing the theoretical understanding and enabling sharper performance guarantees for disk-based search and evacuation problems.

Abstract

We consider \emph{weighted group search on a disk}, which is a search-type problem involving 2 mobile agents with unit-speed. The two agents start collocated and their goal is to reach a (hidden) target at an unknown location and a known distance of exactly 1 (i.e., the search domain is the unit disk). The agents operate in the so-called \emph{wireless} model that allows them instantaneous knowledge of each others findings. The termination cost of agents' trajectories is the worst-case \emph{arithmetic weighted average}, which we quantify by parameter , of the times it takes each agent to reach the target, hence the name of the problem. Our work follows a long line of research in search and evacuation, but quite importantly it is a variation and extension of two well-studied problems, respectively. The known variant is the one in which the search domain is the line, and for which an optimal solution is known. Our problem is also the extension of the so-called \emph{priority evacuation}, which we obtain by setting the weight parameter to . For the latter problem the best upper/lower bound gap known is significant. Our contributions for weighted group search on a disk are threefold. \textit{First}, we derive upper bounds for the entire spectrum of weighted averages . Our algorithms are obtained as a adaptations of known techniques, however the analysis is much more technical. \textit{Second}, our main contribution is the derivation of lower bounds for all weighted averages. This follows from a \emph{novel framework} for proving lower bounds for combinatorial search problems based on linear programming and inspired by metric embedding relaxations. \textit{Third}, we apply our framework to the priority evacuation problem, improving the previously best lower bound known from to , thus reducing the upper/lower bound gap from to .
Paper Structure (14 sections, 11 theorems, 32 equations, 3 figures, 1 table)

This paper contains 14 sections, 11 theorems, 32 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Discussion on New Results
  4. Key Technical and Conceptual Ideas
  5. Roadmap
  6. Preliminaries
  7. Problem Definition, Notation & and some Observations
  8. Past Results, Search Domains & Cost Functions
  9. Improved Framework for Proving Lower Bounds
  10. Implied (and Improved) Lower Bounds for Priority Evacuation
  11. Weighted Group Search on the Disk
  12. Upper Bounds to Weighted Group Search on the Disk
  13. Lower Bounds to Weighted Group Search on the Disk
  14. Conclusion

Key Result

lemma 1

Let $t_0$, $t_1$ be the termination time lower bounds of agents $A_0,A_1$, respectively, for some input to $\textsc{WS}_{n\textsc{-gon}}^{f}$. Then, no algorithm for $\textsc{WS}_{\textsc{disk}}^{f}$ with these termination times of the agents has evacuation cost better than $\frac{ f\left( t_0+\pi/n

Figures (3)

  • Figure 1: Derived upper and lower bounds for $\textsc{WS}_{\textsc{disk}}^{g_w}$, where $g_w(x,y)=wx+y, w\in [0,1]$.
  • Figure 2: The Detour Algorithm for $\textsc{WS}_{\textsc{disk}}^{f}$, and some of its parameters.
  • Figure 3: The behavior of some expressions, as needed in the proof of Lemma \ref{['lem: performance of ab algo for arithmetic']}.

Theorems & Definitions (18)

  • lemma 1
  • proof
  • lemma 2
  • proof
  • corollary 1
  • lemma 3
  • proof
  • lemma 4
  • proof
  • theorem 1
  • ...and 8 more