The maximum independent union of cliques problem: complexity and exact approaches

Given a simple graph, the maximum independent union of cliques problem is to find a maximum-cardinality subset of vertices such that each connected component of the corresponding induced subgraph is a complete graph. This recently introduced problem allows both cliques and independent sets as feasible solutions and is of significant theoretical and applied interest. This paper establishes the complexity of the problem on several classes of graphs (planar, claw-free, and bipartite graphs), and develops an integer programming formulation and an exact combinatorial branch-and-bound algorithm for solving it. Results of numerical experiments with numerous benchmark instances are also reported.