On minimizing vertex bisection using a memetic algorithm

- Vertex Bisection Minimization is an important problem with many applications. In the surveyed literature, no metaheuristic has been reported for the vertex bisection minimization problem. A Memetic Algorithm has been designed for this problem.
- Three construction heuristics have been designed, which are problem dependent, and used for generation of the initial population. These heuristics, in isolation, achieve optimal results for a large number of input instances.
- A crossover and local improvement operator have also been designed and implemented for the Memetic Algorithm.
- Experiments have been conducted to evaluate construction heuristics, analyze the performance of different operators, parameter tuning and to evaluate the performance of the Memetic Algorithm on benchmark graphs.
- We have compared our algorithm with an adapted version of memetic algorithm for the well-known graph partitioning problem. Analysis of the results of the experiments have led to conjectures for some classes of graphs.

Vertex Bisection Minimization problem (VBMP) consists of partitioning a vertex set into two sets B and B′ where |B|=|V|/2 such that vertex width, VW, is minimized which is defined as the number of vertices in B which are adjacent to at least one vertex in B′. It is an NP-complete problem in general. VBMP has applications in fault tolerance and is related to the complexity of sending messages to processors in interconnection networks via vertex disjoint paths. In this paper, a memetic algorithm has been designed for this problem (MAVBMP) in which four construction heuristics have been proposed to generate the initial population. These heuristics are analyzed statistically and accordingly used in proportion to generate the initial population for MAVBMP. A new crossover type search operator has been proposed for recombination and a local improvement operator has also been developed. Extensive experiments have been conducted on several classes of graphs such as complete bipartite graphs, 2-dimensional ordinary meshes, 2-dimensional toroidal meshes, 3-dimensional toroidal meshes, complete split graph, join of hypercubes and Harwell-boeing graphs which are a subset of public domain Matrix Market library. Trends observed in the experimental results for some of the classes of graphs have led to conjectures for vertex width.