Approximating maximum clique with a Hopfield network

A Saturated Linear Dynamical Network for Approximating Maximum Clique

We use a saturated linear gradient dynamical network for finding an approximate solution to the maximum clique problem. We show that for almost all initial conditions, any solution of the network defined on a closed hypercube reaches one of the vertices of the hypercube, and any such vertex corresponds to a maximal clique. We examine the performance of the method on a set of random graphs and c...

The maximum weighted clique problem and Hopfield networks

Nonpositive Hopfield Neural Network with Self-Feedback and its Application to Maximum Clique Problems

A clique of an undirected graph G (V, E) with a vertex set V and an edge set E is a subset of V such that all pairs of vertices are connected by an edge in E. The maximum clique problem (MCP) is to find a clique of maximum size of the graph G. Figure 1(b) shows a maximum clique of the graph Figure 1(a) with 10 vertices and 21 edges. It is one of the first problems which have been proven to be N...

Microcode optimization with neural networks

Microcode optimization is an NP-complete combinatorial optimization problem. This paper proposes a new method based on the Hopfield neural network for optimizing the wordwidth in the control memory of a microprogrammed digital computer. We present two methodologies, viz., the maximum clique approach, and a cost function based method to minimize an objective function. The maximum clique approach...

Approximating the maximum weight clique using replicator dynamics

Given an undirected graph with weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (i.e., a clique) having the largest total weight. This is a generalization of the classical problem of finding the maximum cardinality clique of an unweighted graph, which arises as a special case of the MWCP when all the weights associated to the ve...