**Abstract** : A group of mobile robots (beachcombers) have to search collectively every point of a given domain. At any given moment, each
robot can be in {\em walking mode} or in {\em searching mode}. It is assumed that each robot's maximum allowed searching speed is
strictly smaller than its maximum allowed walking speed. A point of the domain is searched if at least one of the robots visits it
in searching mode. The Beachcombers' Problem consists in developing efficient {\em schedules} (algorithms) for the robots which
collectively search all the points of the given domain as fast as possible.
We first consider the {\em online} Beachcombers' Problem, where the robots are initially collocated at the origin of a
semi-infinite line. It is sought to design a schedule $A$ with maximum {\em speed} $S$, defined as
$
S = \inf_{\ell}{\frac{\ell}{t_A(\ell)}}
$,
where $t_A(\ell)$ denotes the time when the search of the segment $[0,\ell]$ is completed under $A$. We consider a {\em discrete}
and a {\em continuous} version of the problem, depending on whether the infimum is taken over $\ell \in \mathbb{N}^*$ or $\ell
\geq 1$. We prove that the $\mathtt{LeapFrog}$ algorithm, which was proposed in [Czyzowicz et al., SIROCCO 2014, LNCS
8576, pp. 23--36 (2014)], is in fact optimal in the discrete case.
This settles in the affirmative a conjecture from that paper. We also show how to extend this result to the more general
continuous online setting.
For the {\em offline} version of the Beachcombers' Problem, we consider the \emph{single-source} Beachcombers' Problem on the
cycle, as well as the \emph{multi-source} Beachcombers' Problem on the cycle and on the finite segment. For the
\emph{single-source} Beachcombers' Problem on the cycle, we show that the structure of the optimal solutions is identical to the
structure of the optimal solutions to the two-source Beachcombers' Problem on a finite segment. In consequence, by using results
from [Czyzowicz et al., ALGOSENSORS 2014, LNCS 8847, pp. 3--21 (2014)], we prove that the single-source Beachcombers' Problem on
the cycle is NP-hard, and we derive approximation
algorithms for the problem. For the \emph{multi-source} variant of the Beachcombers' Problem on the cycle and on the finite
segment, we obtain efficient approximation algorithms.
One important contribution of our work is that, in all variants of the offline Beachcombers' Problem that we discuss, we allow the
robots to \emph{change direction of movement} and search points of the domain on both sides of their respective starting
positions. This represents a significant generalization compared to the model considered in [Czyzowicz et al., ALGOSENSORS 2014,
LNCS 8847, pp. 3--21 (2014)], in which each robot
had a fixed direction of movement that was specified as part of the solution to the problem. We manage to prove that changes of
direction do not help the robots achieve optimality.