The Max k-Uncut problem arose from the study of homophily of large-scale networks. Given an n-vertex undirected graph $G=(V,E)$ with nonnegative weights defined on edges and a positive integer k, the Max k-Uncut problem asks to find a partition $\{V_1,V_2,\cdots ,V_k\}$ of V such that the total weight of edges that are not cut is maximized. This problem is the complement of the classic Min k-Cut problem, and was proved to have surprisingly rich connection to the Densest k-Subgraph problem. In this paper, we give an approximation algorithm for Max k-Uncut using a non-uniform approach combining LP-rounding and the greedy strategy. The algorithm partitions the vertices of G into at least $(1-\frac{1}{e})k$ parts in expectation, and achieves a good expected approximation ratio $\frac{1}{2}(1+{(\frac{n-k}{n})}^2)$.

在最大 ķ -Uncut问题从同质大规模网络的研究出现。给定一个n -vertex无向图$G=（V，Ë）$在边上定义了非负权重并使用正整数k时，Max k -Uncut问题要求找到一个分区$\{V_{\mathrm{1个}}，V_2，\cdots ，V_{ķ}\}$的V使得边缘是总重量不切断被最大化。该问题是经典Min k- Cut问题的补充，并被证明与Densest k -Subgraph问题具有令人惊讶的丰富联系。在本文中，我们使用结合了LP舍入和贪婪策略的非均匀方法，给出了Max k -Uncut的近似算法。该算法将G的顶点至少划分为$（\mathrm{1个}-\frac{\mathrm{1个}}{Ë}）ķ$ 符合预期，并达到良好的预期近似率 $\frac{\mathrm{1个}}{2}（\mathrm{1个}+{（\frac{ñ-ķ}{ñ}）}^2）$。