Abstract | In 1876 Charles Lutwidge Dodgson suggested the intriguing
voting rule that today bears his name. Although Dodg-
son's rule is one of the most well-studied voting rules, it suf-
fers from serious deciencies, both from the computational
point of view|it is NP-hard even to approximate the Dodg-
son score within sublogarithmic factors|and from the social
choice point of view|it fails basic social choice desiderata
such as monotonicity and homogeneity.
In a previous paper [Caragiannis et al., SODA 2009] we
have asked whether there are approximation algorithms for
Dodgson's rule that are monotonic or homogeneous. In this
paper we give denitive answers to these questions. We de-
sign a monotonic exponential-time algorithm that yields a
2-approximation to the Dodgson score, while matching this
result with a tight lower bound. We also present a monotonic
polynomial-time O(logm)-approximation algorithm (where
m is the number of alternatives); this result is tight as well
due to a complexity-theoretic lower bound. Furthermore,
we show that a slight variation of a known voting rule yields
a monotonic, homogeneous, polynomial-time O(mlogm)-
approximation algorithm, and establish that it is impossible
to achieve a better approximation ratio even if one just asks
for homogeneity. We complete the picture by studying sev-
eral additional social choice properties; for these properties,
we prove that algorithms with an approximation ratio that
depends only on m do not exist. |