  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 $0.25056$.
