  In the Shift Bribery problem, we are given an election (based on preference
orders), a preferred candidate $p$, and a budget. The goal is to ensure that
$p$ wins by shifting $p$ higher in some voters' preference orders. However,
each such shift request comes at a price (depending on the voter and on the
extent of the shift) and we must not exceed the given budget. We study the
parameterized computational complexity of Shift Bribery with respect to a
number of parameters (pertaining to the nature of the solution sought and the
size of the election) and several classes of price functions. When we
parameterize Shift Bribery by the number of affected voters, then for each of
our voting rules (Borda, Maximin, Copeland) the problem is W[2]-hard. If,
instead, we parameterize by the number of positions by which $p$ is shifted in
total,then the problem is fixed-parameter tractable for Borda and Maximin,and
is W[1]-hard for Copeland. If we parameterize by the budget, then the results
depend on the price function class. We also show that Shift Bribery tends to be
tractable when parameterized by the number of voters, but that the results for
the number of candidates are more enigmatic.
