Applying the INN model to the Maximum Clique problem

Max-Clique is the problem of finding the largest clique in a given graph. It is not only NP-hard, but, as recent results suggest, even hard to approximate. Nevertheless it is still very important to develop and test practical algorithms that will find approximate solutions for the maximum clique problem on various graphs stemming from numerous applications. Indeed, many different types of algorithmic approaches are applied to that problem. Several neural networks and related algorithms were applied recently to combinatorial optimization problems in general and to the Max-Clique problem in particular. These neural nets are dynamical system which minimize a cost (or computational ``energy``) function that represents the optimization problem, the Max-Clique in our case. Therefore they all belong to the class of integer programming algorithms surveyed in the Pardalos and Xue review. The work presented here is a development and improvement of a neural network algorithm that was introduced recently. In the previous work, we have considered two Hopfield type neural networks, the INN and the HcN, and their application to the max-clique problem. In this paper, I concentrate on the INN network and present an improved version of the t-A algorithm that was introduced in. The rest of thismore » paper is organized as follows: in section 2, I describe the INN model and how it implements a given graph. In section 3, it is characterized in terms of graph theory. In particular, the stable states of the network are mapped to the maximal cliques of its underling graph. In section 4, I present the t-Annealing algorithm and an improved version of it, the Adaptive t-Annealing. Several experiments done with these algorithms on benchmark graphs are reported in section 5, and the efficiency of the new algorithm is demonstrated. I conclude with a short discussion.« less