  Most theoretical definitions about the complexity of manipulating elections
focus on the decision problem of recognizing which instances can be
successfully manipulated, rather than the search problem of finding the
successful manipulative actions. Since the latter is a far more natural goal
for manipulators, that definitional focus may be misguided if these two
complexities can differ. Our main result is that they probably do differ: If
integer factoring is hard, then for election manipulation, election bribery,
and some types of election control, there are election systems for which
recognizing which instances can be successfully manipulated is in polynomial
time but producing the successful manipulations cannot be done in polynomial
time.
