On the approximability of Dodgson and Young elections

The voting rules proposed by Dodgson and Young are both designed to find an alternative closest to being a Condorcet winner, according to two different notions of proximity; the score of a given alternative is known to be hard to compute under either rule. In this paper, we put forward two algorithms for approximating the Dodgson score: a combinatorial, greedy algorithm and an LP-based algorithm, both of which yield an approximation ratio of Hm- ₁, where m is the number of alternatives and Hm- ₁ is the (m-1)st harmonic number. We also prove that our algorithms are optimal within a factor of 2, unless problems in NP have quasi-polynomial-time algorithms. Despite the intuitive appeal of the greedy algorithm, we argue that the LP-based algorithm has an advantage from a social choice point of view. Further, we demonstrate that computing any reasonable approximation of the ranking produced by Dodgson's rule is NP-hard. This result provides a complexity-theoretic explanation of sharp discrepancies that have been observed in the social choice theory literature when comparing Dodgson elections with simpler voting rules. Finally, we show that the problem of calculating the Young score is NP-hard to approximate by any factor. This leads to an inapproximability result for the Young ranking.