The control complexity of sincere strategy preference based approval voting and of fallback voting, and a study of optimal lobbying and Junta distributions for SAT

While voting systems were originally used in political science, they are now also of central importance in various areas of computer science, such as artificial intelligence (in particular within multiagent systems). Brams and Sanver [BS06] introduced sincere-strategy preference-based approval voting (SP-AV) and fallback voting (FV), two election systems which combine the preference rankings of voters with their approvals of candidates. We study these two systems with respect to procedural control—settings in which an agent seeks to influence the outcome of elections via control actions such as adding/deleting/partitioning either candidates or voters. We prove that SP-AV is computationally resistant (i.e., the corresponding control problems are NP-hard) to 19 out of 22 types of constructive and destructive control. Thus, for the 22 control types studied here, SP-AV has more resistances to control, by three, than is currently known for any other natural voting system with a polynomial-time winner problem. In particular, SP-AV is (after Copeland voting, see Faliszewski et al. [FHHR08a]) the second natural voting system with an easy winner-determination procedure that is known to have full resistance to constructive control, and unlike Copeland voting it in addition displays broad resistance to destructive control. We show that FV has full resistance to candidate control. We also investigate two hard problems related to voting, the optimal weighted lobbying problem and the winner problem for Dodgson elections. Regarding the former problem, Christian et al. [CFRS06] showed that optimal lobbying is intractable in the sense of parameterized complexity. We propose an efficient greedy algorithm that nonetheless approximates a generalized variant of this problem, optimal weighted lobbying, and thus the original optimal lobbying problem as well. We also show that the approximation ratio of this algorithm is tight. The problem of determining Dodgson winners is known to be complete for parallel access to NP [HHR97]. Homan and Hemaspaandra [HH06] proposed an efficient greedy heuristic for finding Dodgson winners with a guaranteed frequency of success, and their heuristic is indeed a “frequently self-knowingly correct algorithm.” We prove that every distributional problem solvable in polynomial time on the average with respect to the uniform distribution has a frequently selfknowingly correct polynomial-time algorithm. Furthermore, we study some features of probability weight of correctness with respect to Procaccia and Rosenschein’s junta distributions [PR07].