Assignment-minimum clique coverings

The search for minimum clique coverings of graphs appears in many practical guises and with several possible minimization goals. One reasonable goal is to minimize the number of overall cliques in a covering, while a second less well-studied but equally reasonable goal is to minimize the number of individual assignments of vertices to cliques. Both goals constitute NP-hard problems and as such require competitive algorithms for practical progress to be made toward their resolutions. In this article, we introduce a technique for accomplishing the latter goal, using a combination of data reduction and a backtracking algorithm. In addition, we demonstrate that it is not always possible to minimize both the number of cliques and the number of individual vertex-clique assignments simultaneously. This demonstration resolves an open question and underscores the need for techniques that specifically minimize the number of assignments of vertices to cliques. We then illustrate our approach in two practical examples. We follow these examples with a simulation-based comparison of our exact approach with a heuristic based on the state-of-the-art algorithm for minimizing the number of cliques in a clique covering. For this comparison, we consider graphs likely to arise in applied statistics, a category of applications for which minimizing individual vertex-clique assignments is of particular interest.