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In 1977, Young proposed a voting scheme that extends the Condorcet Principle based on the fewest possible number of voters whose removal yields a Condorcet winner. We prove that both the winner and the ranking problem for Young elections is complete for P NP , the class of problems solvable in polynomial time by parallel access to NP.

We investigate issues related to two hard problems related to voting, the optimal weighted lobbying problem and the winner problem for Dodgson elections. Regarding the former, Christian et al. [CFRS06] showed that optimal lobbying is intractable in the sense of parameterized complexity. We provide an efficient greedy algorithm that achieves a logarithmic approximation ratio for this problem and even for a more general variant-optimal weighted lobbying. We prove that essentially no better approximation ratio than ours can be proven for this greedy algorithm.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 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 self-knowingly correct polynomial-time algorithm. Furthermore, we study some features of probability weight of correctness with respect to Procaccia and Rosenschein's junta distributions [PR07].Preference aggregation and election systems have been studied for centuries in social choice theory, political science, and economics, see, e.g., Black [Bla58] and McLean and Urken [MU95]. Recently, these topics have become the focus of attention in various areas of computer science as well, such as artificial intelligence (especially with regard to distributed AI in multiagent settings), systems (e.g., for spam filtering), and computational complexity. Faliszewski et al.[FHHR] provides a survey of some recent progress in complexity-related aspects of elections.This paper's topic is motivated by two hard problems that both are related to voting, the optimal weighted lobbying problem and the winner problem for Dodgson elections. Regarding the former problem, Christian et al.[CFRS06] defined its unweighted variant as follows: Given a 0-1 matrix that represents the No/Yes votes for multiple referenda in the context of direct democracy, a positive integer k, and a target vector (of the outcome of the referenda) of an external actor ("The Lobby"), is it possible for The Lobby to reach its target by changing the votes of at most k voters? They proved the optimal lobbying problem complete for the complexity class W[2], thus providing strong evidence that it is intractable even for small values of the parameter k. However, The Lobby might still try to find an approximate solution efficiently. We propose an efficient greedy algorithm that establishes the first approximation result for the weighted version of this problem in which each voter has a price for changing his or her 0-1 vector to The Lobby's specification. Our approximation result applies to Christian et al.'s original optimal lobbying problem (in which each voter has unit price), and also provides the first approximation result for that problem. In particular, we achieve...

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