Solving the bipartite subgraph problems using strictly digital neural networks with virtual slack-neurons

This paper presents a neural network parallel algorithm with SDNN/V (strictly digital neural networks with virtual slacks) enhanced with "virtual slack-neurons" for solving bipartite subgraph problems of combinatorial optimization problems, and the method to improve the quality of solutions by CS (constraint sets) programming based on SDNN/V. This problem is to divide a graph into two clusters so as to minimize the number of removed edges where edges in the same cluster are only removed from the given graph. Note that edges bridging between two clusters are not removed. This problem can be defined as a "set selection problem" with the "between-l-and-k-out-of-n" design rule in SDNN/V algorithm. The number of required neurons to solve this problem using SDNN/V algorithm is V, where V is the number of vertices, and the number of required sets is V+E, where E is the number of edges in a given graph as a bipartite subgraph problem. The 30-vertex with 50-edge graph problem used by other algorithm has been simulated to compare the authors' algorithm with other algorithms. The results of solving the bipartite subgraph problem using the authors' SDNN/V algorithm show that the computation steps in parallel execution is only 2 steps within O(1) time to converge to one of the solutions regardless of the problem size, and that the numbering order of each neuron such as sorted according to the number of sets assigned it has an effect on the quality of solutions in SDNN/V algorithm.