Algorithmic Techniques for Necessary and Possible Winners

We investigate the practical aspects of computing the necessary and possible winners in elections over incomplete voter preferences. In the case of the necessary winners, we show how to implement and accelerate the polynomial-time algorithm of Xia and Conitzer. In the case of the possible winners, where the problem is NP-hard, we give a natural reduction to Integer Linear Programming (ILP) for all positional scoring rules and implement it in a leading commercial optimization solver. Further, we devise optimization techniques to minimize the number of ILP executions and, oftentimes, avoid them altogether. We conduct a thorough experimental study that includes the construction of a rich benchmark of election data based on real and synthetic data. Our findings suggest that, the worst-case intractability of the possible winners notwithstanding, the algorithmic techniques presented here scale well and can be used to compute the possible winners in realistic scenarios

The Complexity of Determining the Necessary and Possible Top-k Winners in Partial Voting Profiles

When voter preferences are known in an incomplete (partial) manner, winner determination is commonly treated as the identification of the necessary and possible winners; these are the candidates who win in all completions or at least one completion, respectively, of the partial voting profile. In the case of a positional scoring rule, the winners are the candidates who receive the maximal total score from the voters. Yet, the outcome of an election might go beyond the absolute winners to the top-$k$ winners, as in the case of committee selection, primaries of political parties, and ranking in recruiting. We investigate the computational complexity of determining the necessary and possible top-$k$ winners over partial voting profiles. Our results apply to general classes of positional scoring rules and focus on the cases where $k$ is given as part of the input and where $k$ is fixed

Computing the Extremal Possible Ranks with Incomplete Preferences

Various voting rules are based on ranking the candidates by scores induced by aggregating voter preferences. A winner (respectively, unique winner) is a candidate who receives a score not smaller than (respectively, strictly greater than) the remaining candidates. Examples of such rules include the positional scoring rules and the Bucklin, Copeland, and Maximin rules. When voter preferences are known in an incomplete manner as partial orders, a candidate can be a possible/necessary winner based on the possibilities of completing the partial votes. Past research has studied in depth the computational problems of determining the possible and necessary winners and unique winners. These problems are all special cases of reasoning about the range of possible positions of a candidate under different tiebreakers. We investigate the complexity of determining this range, and particularly the extremal positions. Among our results, we establish that finding each of the minimal and maximal positions is NP-hard for each of the above rules, including all positional scoring rules, pure or not. Hence, none of the tractable variants of necessary/possible winner determination remain tractable for extremal position determination. Tractability can be retained when reasoning about the top-$k$ positions for a fixed $k$. Yet, exceptional is Maximin where it is tractable to decide whether the maximal rank is $k$ for $k=1$ (necessary winning) but it becomes intractable for all $k>1$.Comment: arXiv admin note: substantial text overlap with arXiv:2002.0921