  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
$\vec{\alpha}=(\alpha_1,\ldots,\alpha_m)$: for some $\beta=O(\sqrt{m\log m})$,
it obtains an additive approximation equal to $W\cdot \max_i \lvert
\alpha_{i+\beta}-\alpha_i \rvert$, 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 $O(k \sqrt{m \log m} )$ approximation; in other words, if
there exists a strategy enabling the preferred candidate to win by an $\Omega(k
\sqrt{m \log m} )$ 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 $\Omega(m)$-additive
approximation to the maximum score of a non-preferred candidate.
  For the weighted case, our generalized algorithm provides an $O(W \sqrt{m
\log m} )$-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.
