A Genetic Algorithm for a Special Class of the Quadratic Assignment Problem

Abstract

A special class of the quadratic assignment problem (QAP) is considered. This class of QAP describes the multiway partitioning problem which is the problem of partitioning a graph into disjoint subgraphs of prescribed sizes by removing the fewest number of edges. A genetic algorithm (GA) for solving this problem is described. A novel feature of this algorithm is the schema preprocessing phase that helps create important building blocks, which in turn improves the performance of the GA. Experimental tests on graphs with published solutions showed that the algorithm performed comparable to or better than the simulated annealing algorithm. Consider the quadratic assignment problem (QAP) where the n x n flow matrix F is a 0-1 symmetric matrix with O'son the main diagonal and the n x n distance matrix D is a block matrix of the form whereObi is a bi x bi matrix of all zeros, for integers b1, ... , bk such that L:~=lbi = n. All other entries of Dare 1. This QAP can be easily seen to describe the followinggraph problem. Let G = (V,E) be an undirected graph on n vertices and k integers b1, ... ,bk be given. The multiway partitioning problem is the 1993 Mathematics Subject Classification. Primary 65KlO; Secondary 68T05, 68RlO. This paper is in preliminary form. problem of finding the smallest set of edges in G whose removal separates G into k disjoint subgraphs Gi = (Vi, Ei), i = 1, ... ,k such that (i) IViI = bi, for all i, (ii) U~=lVi = V, and (iii) Vj n Vi = 0 for j f. t. In other words, it is the problem of finding a partition of the vertex set V into k disjoint subsets of specified sizes and minimizing the number of edges with endpoints in different subsets of the partition. The number of edges having endpoints in different parts of the partition is called the size or cut size of the partition. The flowmatrix F in the QAP is simply the adjacency matrix of the graph G and the number of edges connecting different parts of the partition, Le., the quantity to be minimized, is n L D[i,j)F[1r(i),1r(j)],