  In the computational social choice literature, there has been great interest
in understanding how computational complexity can act as a barrier against
manipulation of elections. Much of this literature, however, makes the
assumption that the voters or agents specify a complete preference ordering
over the set of candidates. There are many multiagent systems applications, and
even real-world elections, where this assumption is not warranted, and this in
turn raises the question "How hard is it to manipulate elections if the agents
reveal only partial preference orderings?" It is this question we try to
address in this paper. In particular, we look at the weighted manipulation
problem -- both constructive and destructive manipulation -- when the voters
are allowed to specify any top-truncated ordering over the set of candidates.
We provide general results for all scoring rules, for elimination versions of
all scoring rules, for the plurality with runoff rule, for a family of election
systems known as Copeland$^{\alpha}$, and for the maximin protocol. Finally, we
also look at the impact on complexity of manipulation when there is uncertainty
about the non-manipulators' votes.
