  We give the first quasipolynomial upper bound $\phi n^{\text{polylog}(n)}$
for the smoothed complexity of the SWAP algorithm for local Graph Partitioning
(also known as Bisection Width), where $n$ is the number of nodes in the graph
and $\phi$ is a parameter that measures the magnitude of perturbations applied
on its edge weights. More generally, we show that the same quasipolynomial
upper bound holds for the smoothed complexity of the 2-FLIP algorithm for any
binary Maximum Constraint Satisfaction Problem, including local Max-Cut, for
which similar bounds were only known for $1$-FLIP. Our results are based on an
analysis of cycles formed in long sequences of double flips, showing that it is
unlikely for every move in a long sequence to incur a positive but small
improvement in the cut weight.
