A Different Approach to Maximum Clique Search

The way we tackle NP-hard problems in practical setting has experienced a major shift in recent years. Our view has became more sophisticated with the emergence of the parameterized complexity paradigm. We may distinguish subclasses inside the NP-hard complexity class. The complexity of the problems in different subclasses maybe quite different. The overall conservative estimate of the running time is replaced by a more optimistic estimate. In addition the approach of parameterized algorithms is sometimes able to deal with the more complex problems by dividing the problem into harder and a simpler parts. The easier instance at many times reduces to mere preprocessing step leaving us with only the harder part. In this paper we single out the so-called maximum clique problem as a typical representative of the NP-hard complexity class. We propose an algorithm to solve the maximum clique problem motivated by the above ideas. Many of the available maximum clique solvers are descendants or refined versions of the Carraghan–Pardalos algorithm. (Patric \"Osterg\aa rd's cliquer is being as an exception.) The maximum clique problem as a maximization problem can be reduced to a series of k-clique problems as decision problems. Our main observation is that this route offers a number of advantages. The structure of a k-clique decision problem is simpler than the structure of a maximization problem. It affords additional pruning opportunities based on the available value of k. A large scale numerical experiment indicates that in many occasions the combined search space of the k-clique problems is smaller than the search space of the maximization problem. The solver we propose turns out to be rather efficient. In a number of test problems it beats the best available solvers.