  A set of mobile robots (represented as points) is distributed in the
Cartesian plane. The collection contains an unknown subset of byzantine robots
which are indistinguishable from the reliable ones. The reliable robots need to
gather, i.e., arrive to a configuration in which at the same time, all of them
occupy the same point on the plane. The robots are equipped with GPS devices
and at the beginning of the gathering process they communicate the Cartesian
coordinates of their respective positions to the central authority. On the
basis of this information, without the knowledge of which robots are faulty,
the central authority designs a trajectory for every robot. The central
authority aims to provide the trajectories which result in the shortest
possible gathering time of the healthy robots. The efficiency of a gathering
strategy is measured by its competitive ratio, i.e., the maximal ratio between
the time required for gathering achieved by the given trajectories and the
optimal time required for gathering in the offline case, i.e., when the faulty
robots are known to the central authority in advance. The role of the byzantine
robots, controlled by the adversary, is to act so that the gathering is delayed
and the resulting competitive ratio is maximized.
  The objective of our paper is to propose efficient algorithms when the
central authority is aware of an upper bound on the number of byzantine robots.
We give optimal algorithms for collections of robots known to contain at most
one faulty robot. When the proportion of byzantine robots is known to be less
than one half or one third, we provide algorithms with small constant
competitive ratios. We also propose algorithms with bounded competitive ratio
in the case where the proportion of faulty robots is arbitrary.
