We study the problem of coalitional manipulation---where k manipulators try to manipulate an election on m candidates---for any scoring rule, with focus on the Borda protocol. We do so in both the weighted and unweighted settings. For these problems, recent approximation approaches have tried to minimize k, the number of manipulators needed to make some preferred candidate p win (thus assuming that the number of manipulators is not limited in advance). In contrast, we focus on minimizing the score margin of p which is the difference between the maximum score of a candidate and the score of p.
We provide algorithms that approximate the optimum score margin, which are applicable to any scoring rule. For the specific case of the Borda protocol in the unweighted setting, our algorithm provides a superior approximation factor for lower values of k.
Our methods are novel and adapt techniques from multiprocessor scheduling by carefully rounding an exponentially-large configuration linear program that is solved by using the ellipsoid method with an efficient separation oracle. We believe that such methods could be beneficial in other social choice settings as well.