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

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 \(\mathsf{Max}\;k\text {-}\mathsf{Uncut}\).