Approximation and Hardness Results for the Max k-Uncut Problem

Abstract

In this paper, we propose the \(\mathsf{Max}~k\text {-}\mathsf{Uncut}\) problem. Given an n-vertex undirected graph \(G = (V, E)\) with nonnegative weights \(\{w_e \mid e \in E\}\) 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. This problem is just the complement of the classic \(\mathsf{Min}~k\text {-}\mathsf{Cut}\) problem. We get this problem from the study of complex networks. For \(\mathsf{Max}~k\text {-}\mathsf{Uncut}\), we present a randomized \((1-\frac{k}{n})^2\)-approximation algorithm, a greedy \((1-\frac{2(k-1)}{n})\)-approximation algorithm, and an \(\varOmega (\frac{1}{2} \alpha )\)-approximation algorithm by reducing it to \(\mathsf{Densest}~k\text {-}\mathsf{Subgraph}\), where \(\alpha \) is the approximation ratio for the \(\mathsf{Densest}~k\text {-}\mathsf{Subgraph}\) problem. More importantly, we show that \(\mathsf{Max}~k\text {-}\mathsf{Uncut}\) and \(\mathsf{Densest}~k\text {-}\mathsf{Subgraph}\) are in fact equivalent in approximability up to a factor of 2. We also prove a weak approximation hardness result for \(\mathsf{Max}~k\text {-}\mathsf{Uncut}\) under the assumption \(\text {P} \ne \text {NP}\).