Graph Coloring with the Hop eld-clique Network

We approximately solve, by reduction to Maximum Clique, the graph k-coloring NP-hard problem in a binary weights Hoppeld net special case. This network was used earlier to approximately solve Maximum Clique and some other NP-hard problems by reduction to Maximum Clique. We determine k-coloring approximation performance on random graphs and on one other distribution of \harder" graphs. We compare our work with earlier work and with theoretical estimates on random graphs. Our optimizing dynamics are discrete and converge in O(number of units) unit-switches. Our broad contribution is in optimizing a new problem in the same binary weights (0/-1) network, including the same dynamics, employed earlier for other problems. Our mapping does not admit any invalid solutions and is goodness-preserving in a formal sense. We also view the act of loading problem instances to be solved as associative memories storage. MAXIMUM CLIQUE. In a graph with undirected edges, a clique is a set of vertices such that every pair is connected by an edge. A clique is maximal if no strict superset of it is also a clique. A clique is maximum if it is the largest clique. A k-clique is a clique of size k. are not clique, non-maximal 3-clique, maximal-but-not-maximum clique, and maximum clique (size 4) respectively. For neural net readers, maximal (or minimal) and maximum (or minimum) correspond to local and global optima respectively. MAX-CLIQUE is the optimization problem of nding a largest clique in a given graph and is NP-hard 22] even to approximate 4]. In a recent paper, we have approximately solved some NP-hard problems by reduction to MAX-CLIQUE followed by approximate solution of MAX-CLIQUE in a Hoppeld network special case (HcN) whose stable states are maximal cliques 20] (also see 18]). In the current work, we add a new problem, solved in a similar way, to this list: graph k-coloring. The reduction of graph k-coloring to MAX-CLIQUE described here has the following property: every maximal clique in a reduced instance corresponds to some feasible solution of the original problem. Since HcN admits exactly the maximal cliques as stable states (see Lemma 1 later), no invalid solutions of a reduced problem are thus admitted as stable states. In the terminology of 1] (page 154), our mapping of graph k-coloring to HcN is feasible. Problems of poor optimization by neural networks in the past can be attributed to mappings that are not feasible (see 15, 3]). Deene …