Multilevel + Neural Network Heuristic for the Graph Bisection Problem on Geometrically Connected Graphs

Abstract

The Multilevel algorithm (ML) has been applied successfully as a metaheuristic for different combinatorial optimization problems: Graph Partitioning, Traveling Salesman, Graph Coloring, see refs. [6,7,18]. The main difficulty of ML are the convergence times needed to obtain solutions at a distance of 7% - 5% to the best known solution in large scale problems. In order to reduce these convergence times we studied numerically a Parallel Multilevel heuristic with Neural Network partitioning and uncoarsening + refinement phases (PML+PNN) for the Graph Bisection Problem on geometrically connected graphs. Our main result establish that for graphs with n∊[4000,12000] vertices, the performance of the parallel ML+NN heuristic increases linearly as n increases with respect to the parallel ML heuristic. For n∊{10000,12000} the distance to the best solution found is 0.32,0.25 respectively that is obtained with a quadratic computing time. This suggests improving the performance of the PML+PNN heuristic by means of a hill climbing improvement heuristic.