Solving the Maximum Clique Problem with Symmetric Rank-One Non-negative Matrix Approximation

Finding complete subgraphs in a graph, that is, cliques, is a key problem and has many real-world applications, e.g., finding communities in social networks, clustering gene expression data, modeling ecological niches in food webs, and describing chemicals in a substance. The problem of finding the largest clique in a graph is a well-known difficult combinatorial optimization problem and is called the maximum clique problem. In this paper, we formulate a very convenient continuous characterization of the maximum clique problem based on the symmetric rank-one non-negative approximation of a given matrix and build a one-to-one correspondence between stationary points of our formulation and cliques of a given graph. In particular, we show that the local (resp. global) minima of the continuous problem corresponds to the maximal (resp. maximum) cliques of the given graph. We also propose a new and efficient clique finding algorithm based on our continuous formulation and test it on the DIMACS data sets to show that the new algorithm outperforms other existing algorithms based on the Motzkin–Straus formulation and can compete with a sophisticated combinatorial heuristic.