  We study the problem of bribery in multiwinner elections, for the case where
the voters cast approval ballots (i.e., sets of candidates they approve) and
the bribery actions are limited to: adding an approval to a vote, deleting an
approval from a vote, or moving an approval within a vote from one candidate to
the other. We consider a number of approval-based multiwinner rules (AV, SAV,
GAV, RAV, approval-based Chamberlin--Courant, and PAV). We find the landscape
of complexity results quite rich, going from polynomial-time algorithms through
NP-hardness with constant-factor approximations, to outright inapproximability.
Moreover, in general, our problems tend to be easier when we limit out bribery
actions on increasing the number of approvals of the candidate that we want to
be in a winning committee (i.e., adding approvals only for this preferred
candidate, or moving approvals only to him or her). We also study parameterized
complexity of our problems, with a focus on parameterizations by the numbers of
voters or candidates.
