Approximating Max k-Uncut via LP-rounding Plus Greed, with Applications to Densest k-Subgraph

Abstract

The $\mathsf{Max}\;k\text {-}\mathsf{Uncut}$ 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 $\mathsf{Max}\;k\text {-}\mathsf{Uncut}$ 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. $\mathsf{Max}\;k\text {-}\mathsf{Uncut}$ can also be viewed as a clustering problem with the measure being the total weight of uncut edges in the solution. This problem is the complement of the classic $\mathsf{Min}\;k\text {-}\mathsf{Cut}$ problem, and was proved to have surprisingly rich connection to the Densest $k\text {-}\mathsf{Subgraph}$ problem. In this paper, we give approximation algorithms for $\mathsf{Max}\;k\text {-}\mathsf{Uncut}$ using a non-uniform approach combining LP-rounding and the greedy strategy. With a limited violation of the constraint k, we present a good expected approximation ratio $\frac{1}{2}(1+(\frac{n-k}{n})^2)$ for $\mathsf{Max}\;k\text {-}\mathsf{Uncut}$.