A Multilevel Memetic Approach for Improving Graph k-Partitions

Abstract

Graph partitioning is one of the most studied NP-complete problems. Given a graph <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</i> =( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> ) , the task is to partition the vertex set <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">V</i> into <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> disjoint subsets of about the same size, such that the number of edges with endpoints in different subsets is minimized. In this paper, we present a highly effective multilevel memetic algorithm, which integrates a new multiparent crossover operator and a powerful perturbation-based tabu search algorithm. The proposed crossover operator tends to preserve the backbone with respect to a certain number of parent individuals, i.e., the grouping of vertices which is common to all parent individuals. Extensive experimental studies on numerous benchmark instances from the graph partitioning archive show that the proposed approach, within a time limit ranging from several minutes to several hours, performs far better than any of the existing graph partitioning algorithms in terms of solution quality.