We consider the semi-random graph model of [Makarychev, Makarychev and Vijayaraghavan, STOC'12], where, given a random bipartite graph with $\alpha$ edges and an unknown bipartition $(A, B)$ of the vertex set, an adversary can add arbitrary edges inside each community and remove arbitrary edges from the cut $(A, B)$ (i.e. all adversarial changes are \textit{monotone} with respect to the bipartition). For this model, a polynomial time algorithm is known to approximate the Balanced Cut problem up to value $O(\alpha)$ [MMV'12] as long as the cut $(A, B)$ has size $\Omega(\alpha)$. However, it consists of slow subroutines requiring optimal solutions for logarithmically many semidefinite programs. We study the fine-grained complexity of the problem and present the first near-linear time algorithm that achieves similar performances to that of [MMV'12]. Our algorithm runs in time $O(|V(G)|^{1+o(1)} + |E(G)|^{1+o(1)})$ and finds a balanced cut of value $O(\alpha)$. Our approach appears easily extendible to related problem, such as Sparsest Cut, and also yields an near-linear time $O(1)$-approximation to Dagupta's objective function for hierarchical clustering [Dasgupta, STOC'16] for the semi-random hierarchical stochastic block model inputs of [Cohen-Addad, Kanade, Mallmann-Trenn, Mathieu, JACM'19].

翻译：我们考虑[Makarychev, Makarychev和Vijayaraghavan, STOC'12]提出的半随机图模型：给定一个具有$\alpha$条边的随机二分图及顶点集未知的二分划$(A, B)$，对手可以在每个社区内部任意添加边，并从割$(A, B)$中任意删除边（即所有对抗性修改相对于二分划都是\textit{单调的}）。针对该模型，已知存在多项式时间算法可将平衡割问题近似至$O(\alpha)$值[MMV'12]，前提是割$(A, B)$的规模为$\Omega(\alpha)$。然而，该算法包含缓慢的子程序，需要对数数量的半定规划进行最优求解。我们研究该问题的精细复杂度，提出了首个达到与[MMV'12]相似性能的近线性时间算法。我们的算法运行时间为$O(|V(G)|^{1+o(1)} + |E(G)|^{1+o(1)})$，并能找到值为$O(\alpha)$的平衡割。我们的方法似乎能轻松扩展到相关问题，如稀疏割，同时针对[Cohen-Addad, Kanade, Mallmann-Trenn, Mathieu, JACM'19]提出的半随机分层随机块模型输入，还为Dasgupta层次聚类目标函数[Dasgupta, STOC'16]提供了近线性时间的$O(1)$近似解。