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 $0.25056$.

翻译：本文研究\emph{圆盘上的加权群搜索问题}，这是一个涉及两个单位速度移动智能体的搜索类问题。两个智能体从同一位置出发，目标是到达位于未知位置但已知距离恰好为1的（隐藏）目标点（即搜索区域为单位圆盘）。智能体在所谓的\emph{无线}模型下运作，该模型允许它们即时获知彼此的发现结果。智能体轨迹的终止代价是两者到达目标所需时间的\emph{算术加权平均}的最坏情况值，我们通过参数$w$来量化该权重，问题由此得名。我们的工作遵循搜索与撤离领域的一系列长期研究，但尤为重要的是，它分别是两个被深入研究的变体与扩展问题。已知的变体是搜索区域为直线的情况，其最优解已获知。我们的问题也是所谓\emph{优先级撤离}问题的扩展——通过将权重参数$w$设为0即可得到后者。对于优先级撤离问题，目前已知的最优上下界差距显著。我们在圆盘加权群搜索问题上的贡献有三方面。\textit{首先}，我们推导了全权重范围$w$下的上界。所得算法是对已知技术的改进，但其分析过程更具技术挑战性。\textit{其次}，我们的主要贡献是针对所有权重值推导出下界。这源于一种\emph{新颖的证明框架}，该框架基于线性规划并受度量嵌入松弛思想的启发，用于证明组合搜索问题的下界。\textit{第三}，我们将该框架应用于优先级撤离问题，将先前最佳下界从$4.38962$提升至$4.56798$，从而将上下界差距从$0.42892$缩减至$0.25056$。