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.

翻译：我们首次给出了局部图划分（也称为二分宽度）中SWAP算法的平滑复杂度的准多项式上界 $\phi n^{\text{polylog}(n)}$，其中 $n$ 是图中的节点数，$\phi$ 是衡量其边权重上施加扰动幅度的参数。更一般地，我们证明对于任意二元最大约束满足问题（包括局部最大割，此前类似上界仅对 $1$-FLIP 成立），其 $2$-FLIP 算法的平滑复杂度同样满足该准多项式上界。我们的结果基于对长序列双翻转中形成的循环的分析，表明在长序列中每一步翻转都产生正但微小割权重改进的可能性很低。