Approximation and hardness results for the Max k-Uncut problem

Abstract In the study of the homophily law of large scale complex networks, we get a combinatorial optimization problem which we call the Max k -Uncut problem. Given an n -vertex undirected graph G = ( V , E ) with nonnegative weights { w e | e ∈ E } defined on edges, and a positive integer k , the Max k -Uncut problem asks to find a partition { V 1 , V 2 , ⋯ , V k } of V such that the total weight of edges that are not cut is maximized. Intuitively, an edge that is not cut connects two vertices with the same or similar attributes since they are in the same part of the partition. Interestingly, the Max k -Uncut problem is just the complement of the classic Min k -Cut problem. For Max k -Uncut , we present a randomized ( 1 − k n ) 2 -approximation algorithm, a greedy ( 1 − 2 ( k − 1 ) n ) -approximation algorithm, and an Ω ( 1 2 α ) -approximation algorithm by reducing it to Densest k -Subgraph , where α is the approximation ratio of the Densest k -Subgraph problem. More importantly, we show that Max k -Uncut and Densest k -Subgraph are in fact equivalent in approximability up to a factor of 2. We also prove an approximation hardness result for Max k -Uncut under the assumption P ≠ NP .