Approval Voting: Local Search Heuristics and Approximation Algorithms for the Minimax Solution

Voting has been the most general scheme for preference aggregation in multi-agent settings involving agents of diverse preferences. Here, we study a specific type of voting protocols for multi-winner elections, namely approval voting, and we investigate the complexity of computing or approximating the minimax solution in approval voting, concentrating on elections for committees of fixed size. Given an approval voting election, where voters can vote for as many candidates as they like, a minimax outcome is a committee that minimizes the maximum Hamming distance to the voters’ ballots. We first show that the problem is NP-hard and give a simple 3-approximation algorithm. We then introduce and evaluate various heuristics based on local search. Our heuristics have low running times (and provably polynomial) and our experimental results show that they perform very well on average, computing solutions that are very close to the optimal minimax solutions. Finally, we address the issue of manipulating minimax outcomes. We show that even though exact algorithms for the minimax solution are manipulable, we can have approximation algorithms that are non-manipulable.