  We study some basic algorithmic problems concerning the intersection of
tropical hypersurfaces in general dimension: deciding whether this intersection
is nonempty, whether it is a tropical variety, and whether it is connected, as
well as counting the number of connected components. We characterize the
borderline between tractable and hard computations by proving
$\mathcal{NP}$-hardness and #$\mathcal{P}$-hardness results under various
strong restrictions of the input data, as well as providing polynomial time
algorithms for various other restrictions.
