A Quadratic Integer Programming with Application in Chaotic Mappings of Complete Multipartite Graphs

Abstract

Let alpha be a permutation of V(G) of a connected graph G. Define the total relative displacement of alpha in G by

where dG(x,y) is the length of the shortest path between x and y in G. Let pi*(G) be the maximum value of deltaalpha(G) among all permutations of V(G) and the permutation which realizes pi*(G) is called a chaotic mapping of G. In this paper, we study the chaotic mappings of complete multipartite graphs. The problem will reduce to a quadratic integer programming. We characterize its optimal solution and present an algorithm running in O(n5log n) time where n is the total number of vertices in a complete multipartite graph.