On the exact amount of missing information that makes finding possible winners hard

In the possible winner problem, we need to compute if a set of partial votes can be completed such that a given candidate wins the election under some specific voting rule. In this paper, we determine the smallest number of undetermined pairs per partial vote for which the Possible Winner problem is $\mathrm{NP}$-complete. In particular, we find the exact values of t for which the Possible Winner problem transitions to being $\mathrm{NP}$-complete from being in $P$, where t is the maximum number of undetermined pairs in every vote. We demonstrate tight results for a broad class of scoring rules, Copeland^α for every $\alpha \in [0,1]$, maximin, and Bucklin voting rules. A somewhat surprising aspect of our results is that for many of these rules, the Possible Winner problem turns out to be hard even if every vote has at most one undetermined pair of candidates.