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 $nk/4+n{P}_{\star }\sqrt{k/4}+\mathsf{err}(n,k,L)$, where $P_{\star }\approx 0.763166$ is the value of the Parisi formula from spin glass theory, and $\mathsf{err}(n,k,L)=o_n(n)+n{o}_k(\sqrt{k})+n\sqrt{k}{o}_L(1)$ (subscripts indicate the asymptotic variables). Our result generalizes to locally treelike graphs, that is, 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 $nk/4+n{P}_{\star }\sqrt{k/4}+o_n(n)+n{o}_k(\sqrt{k})$. 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.

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大度局部树状正则图的最大割和最小二分局部算法

给定一个图$G$学位的$k$超过$n$顶点，我们考虑在多项式时间内计算接近最大割或接近最小平分的问题。对于周长图表$2L$，我们开发了一种本地消息传递算法，其复杂度为$氧（nkL）$，并且在所有参数中实现了接近最佳的切割值$L$-本地算法。算法以max-cut为重点，构造价值的割$nk/4+n{磷}_{\star }\sqrt{k/4}+\mathsf{呃}（n,k,L）$， 在哪里${磷}_{\star }\approx 0。763166$是自旋玻璃理论的 Parisi 公式的值，并且$\mathsf{呃}（n,k,L）={哦}_n（n）+n{哦}_k（\sqrt{k}）+n\sqrt{k}{哦}_L（1）$（下标表示渐近变量）。我们的结果推广到局部树状图，即周长变为$2L$删除一小部分顶点后。早期的工作表明，对于随机$k$- 常规图，典型的最大割值为$nk/4+n{磷}_{\star }\sqrt{k/4}+{哦}_n（n）+n{哦}_k（\sqrt{k}）$。因此，我们的算法在此类图上几乎是最优的。这个结果的直接推论是，随机正则图在所有正则局部树状图中具有接近最小最大割和接近最大最小二分。这可以被视为随机正则图的近拉马努金性质的组合版本。