New Approximations for Coalitional Manipulation in General Scoring Rules,

We study the problem of coalitional manipulation---where k manipulators try to manipulate an election on m candidates---under general scoring rules, with a focus on the Borda protocol. We do so both in the weighted and unweighted settings. We focus on minimizing the maximum score obtainable by a non-preferred candidate.
In the strongest, most general setting, we provide an algorithm for any scoring rule as described by a vector α⃗ =(α1,…,αm): for some β=O(mlogm−−−−−−√), it obtains an additive approximation equal to W⋅maxi|αi+β−αi|, where W is the sum of voter weights.
For Borda, both the weighted and unweighted variants are known to be NP-hard. For the unweighted case, our simpler algorithm provides a randomized, additive O(kmlogm−−−−−−√) approximation; in other words, if there exists a strategy enabling the preferred candidate to win by an Ω(kmlogm−−−−−−√) margin, our method, with high probability, will find a strategy enabling her to win (albeit with a possibly smaller margin). It thus provides a somewhat stronger guarantee compared to the previous methods, which implicitly implied a strategy that provides an Ω(m)-additive approximation to the maximum score of a non-preferred candidate.
For the weighted case, our generalized algorithm provides an O(Wmlogm−−−−−−√)-additive approximation, where W is the sum of voter weights. This is a clear advantage over previous methods: some of them do not generalize to the weighted case, while others---which approximate the number of manipulators---pose restrictions on the weights of extra manipulators added.
Our methods are based on carefully rounding an exponentially-large configuration linear program that is solved by using the ellipsoid method with an efficient separation oracle.