An Incremental Genetic Algorithm for Graph Optimization Problems

Abstract

A combinatorial optimization problem is an optimization problem having a discrete solution space. Lots of the graph problems belong to this category as graphs are discrete objects. Graphs are widely used in the various field and there are lots of real world combinatorial optimization problems which take the graphs as their input. For some of these problems, the magnitude of the solution space is exponential to the size of the problem, and thereby efficient space search algorithms are required to deal with them.

Genetic algorithms are widely used to solve combinatorial optimization problems, and incremental genetic algorithms could be used to efficiently solve graph optimization problems.We define subproblems and solve them step by step instead of tackling the problems directly. A subproblem solved by an incremental genetic algorithm deals with a restriction of the original graph structure. The subproblems are solved in the intermediate steps and the size of the subproblem is gradually increased. We apply the same genetic algorithm to each subproblem, and it is initialized with the evolved population of the previous step.

We propose incremental genetic algorithms for two different combinatorial optimization problems

the subgraph isomorphism problem and graph cut optimization problem. We devise an optimal substructure on the subproblem sequence and explain how it is related to the optimality of the process, along with other related factors. We present graph expansion methodologies and vertex reordering schemes to define an appropriate sequence of subproblems. We combine the proposed incremental approach with a hybrid genetic algorithm for the subgraph isomorphism problem, and the algorithm was further developed for nearly perfect results. Based on our analysis, we also propose an incremental genetic algorithm to solve graph cut optimization problems. We tested the implementation of the algorithm on benchmark graph instances for the graph partitioning problem and the maximum cut problem. Through experiments, we investigate and analyze how the sequence of subproblems affects the search space landscape. The performance of a genetic algorithm makes an improvement when the incremental approach is applied with respect to an appropriate sequence of subproblems.