Given a graph $G$ of degree $k$ over $n$ vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth $2L$, we develop a local message passing algorithm whose complexity is $O(nkL)$, and that achieves near optimal cut values among all $L$-local algorithms. Focusing on max-cut, the algorithm constructs a cut of value $nk/4+ n\mathsf{P}_\star\sqrt{k/4}+\mathsf{err}(n,k,L)$, where $\mathsf{P}_\star\approx 0.763166$ is the value of the Parisi formula from spin glass theory, and $\mathsf{err}(n,k,L)=o_n(n)+no_k(\sqrt{k})+n \sqrt{k} o_L(1)$ (subscripts indicate the asymptotic variables). Our result generalizes to locally treelike graphs, i.e., graphs whose girth becomes $2L$ after removing a small fraction of vertices. Earlier work established that, for random $k$-regular graphs, the typical max-cut value is $nk/4+ n\mathsf{P}_\star\sqrt{k/4}+o_n(n)+no_k(\sqrt{k})$. Therefore our algorithm is nearly optimal on such graphs. An immediate corollary of this result is that random regular graphs have nearly minimum max-cut, and nearly maximum min-bisection among all regular locally treelike graphs. This can be viewed as a combinatorial version of the near-Ramanujan property of random regular graphs.

翻译：给定一个$n$个顶点、度为$k$的图$G$，我们考虑在多项式时间内计算近优最大割或近优最小二分划的问题。对于围长为$2L$的图，我们开发了一种局部消息传递算法，其复杂度为$O(nkL)$，并且能在所有$L$局部算法中实现近优割值。以最大割为例，该算法构造的割值为$nk/4+ n\mathsf{P}_\star\sqrt{k/4}+\mathsf{err}(n,k,L)$，其中$\mathsf{P}_\star\approx 0.763166$是自旋玻璃理论中Parisi公式的值，而$\mathsf{err}(n,k,L)=o_n(n)+no_k(\sqrt{k})+n \sqrt{k} o_L(1)$（下标表示渐近变量）。我们的结果推广到局部树状图，即移除一小部分顶点后围长变为$2L$的图。先前的工作表明，对于随机$k$-正则图，典型最大割值为$nk/4+ n\mathsf{P}_\star\sqrt{k/4}+o_n(n)+no_k(\sqrt{k})$。因此，我们的算法在此类图上近乎最优。该结果的一个直接推论是：在所有正则局部树状图中，随机正则图拥有近最小的最大割和近最大的最小二分划。这可视为随机正则图近拉马努金性质的组合版本。