A New Global Algorithm for Max-Cut Problem with Chordal Sparsity

In this paper, we develop a semidefinite relaxation-based branch-and-bound algorithm that exploits the chordal sparsity patterns of the max-cut problem. We first study how the chordal sparsity pattern affects the hardness of a max-cut problem. To do this, we derive a polyhedral relaxation based on the clique decomposition of the chordal sparsity patterns and prove some sufficient conditions for the tightness of this polyhedral relaxation. The theoretical results show that the max-cut problem is easy to solve when the sparsity pattern embedded in the problem has a small treewidth and the number of vertices in the intersection of maximal cliques is small. Based on the theoretical results, we propose a new branching rule called hierarchy branching rule, which utilizes the tree decomposition of the sparsity patterns. We also analyze how the proposed branching rule affects the chordal sparsity patterns embedded in the problem, and explain why it can be effective. The numerical experiments show that the proposed algorithm is superior to those known algorithms using classical branching rules and the state-of-the-art solver BiqCrunch on most instances with sparsity patterns arisen in practical applications.