Computational Complexity in Three Areas of Computational Social Choice: Possible Winners, Unidirectional Covering Sets, and Judgment Aggregation

This thesis studies the computational complexity of different problems from three areas of computational social choice. The first one is voting, and especially the problem of determining whether a distinguished candidate can be a winner in an election with some kind of incomplete information. The second setting is in the broader sense related to the problem of determining winners. Here the computational complexity of problems related to minimal upward and downward covering sets is investigated. The last area is judgment aggregation. In contrast to the problems mentioned above we do not study the complexity of some kind of “winner” problem, but the complexity of three forms of influencing the outcome, namely manipulation, bribery, and control. All studied problems come from computational social choice, which is a field at the interface between social choice theory and computer science, with a bidirectional transfer between these two disciplines. We focus on the study of the computational complexity of problems coming from social choice theory. One central problem in social choice is that of winner determination in elections. From a computational point of view it is desirable that the winner can be determined in polynomial time. Associated with this problem is the possible winner problem. Here, the question is whether an election, which is in some sense incompletely specified, can be completed such that a distinguished candidate wins the election. In contrast to the winner problem, it is not always desirable that possible winners can be computed in polynomial time, since this may give incentive to some kind of manipulation in the voting process. The first part of the thesis deals with the complexity of different possible winner problems, and establishes results for the classes P and NP. Also related to the winner problem in voting are solution concepts for dominance graphs as they may result from a pairwise majority relation. A solution concept is a way of identifying the “most desirable” elements of such dominance graphs. In the second part of this thesis, we study the complexity of various problems related to so-called upward and downward covering sets. We show hardness and completeness not only for NP, but also for the complexity classes coNP and Θp2, and we show membership in Σ p 2. The last part of this thesis is concerned with judgment aggregation. Here the task is not to determine a winner, but to aggregate the individual judgment sets over possibly interconnected logical propositions. We study manipulation, bribery, and control in such processes. The manipulation problem asks whether a judge has an incentive to report an untruthful judgment set, in the bribery problem an external actor seeks to change the outcome by bribing some of the judges, and in the control problems the set of participating judges may be changed. Again, this may be undesirable, hence showing NPhardness can be seen as providing some kind of protection against manipulation, bribery, and control. In addition to classical complexity results, we also study the parameterized complexity and establish W[2]-hardness for various problems.