A maximum hypergraph 3-cut problem with limited unbalance: approximation and analysis - Sun, Jian; Zhang, Zan-Bo; Chen, Yannan; Han, Deren; Du, Donglei; Zhang, Xiaoyan (Journal of global optimization, 2023)
We consider the max hypergraph 3-cut problem with limited unbalance (MH3C-LU). The objective is to divide the vertex set of an edge-weighted hypergraph H = ( V , E , w ) into three disjoint subsets V 1 , V 2 , and V 3 such that the sum of edge weights cross different parts is maximized subject to | | V i | - | V l | | ≤ B ( ∀ i ≠ l ∈ { 1 , 2 , 3 } ) for a given parameter B . This problem is NP-hard because it includes some well-known problems like the max 3-section problem and the max 3-cut problem as special cases. We formulate the MH3C-LU as a ternary quadratic program and present a randomized approximation algorithm based on the complex semidefinite programming relaxation technique.