Quadratic Integer Programming with Application to the Chaotic Mappings of Complete Multipartite Graphs

Abstract

Let be a permutation of V (G) of a connected graph G. De ne the total relativ e displacement of in G by (G) = X x;y2V (G) jd G (x; y) ? d G ((x); (y))j where d G (x; y) is the length of the shortest path between x and y in G. Let (G) be the maximum value of (G) among all permutations of V (G) and the permutation which realizes (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(n 5 log n) time where n is the total number of vertices in a complete multipartite graph. key word. Chaotic mapping, complete multipartite graph. AMS(MOS) subject classi cation. 05C05

Extending the QCR method to general mixed-integer programs

“…Some of these applications can be formulated as (M QP ). For instance, (M QP ) is used in [10] for the unit commitment problem and for the Markowitz mean-variance model, in [11] for the chaotic mapping of complete multipartite graphs, in [7] for the material cutting, and in [17] for the capacity planning.…”

The Chaotic Numbers of the Bipartite and Tripartite Graphs

“…al. [5] transformed the problem of finding π * (K n1,n2,···,nt ) into the following equivalent quadratic integer programming:…”

“…On the other hand, Fu et al [5] and Chiang and Tzeng [4] studied the maximum value of the total relative displacements of permutations in a graph G, denoted by π * (G) and called by the chaotic number of G, and the chaotic mapping of the graph G which attains π * (G). In fact, on the application point of view, π * (G) is much worthy to study.…”

“…It measures the complexity of the disorderedness of a data structure in the worse case [6]. In [5], the problem of finding π * (K n1,n2,···,nt ) was transformed into a quadratic integer programming problem, and a characterization of the optimal solution was given and then they gave a polynomial time algorithm running in O(n 5 log n) time to solve the problem where n is the number of vertices in the t-partite graph. Later Chiang and Tzeng [4] studied total self-variations of sequences and use this concept to develop a best possible upper bound for π * (G) of a graph G.…”

“…In this paper, we study the total relative displacement of permutations in a bipartite graph and a tripartite graph along an approashing different from that in [5]. We find the closed form of the chaotic number for bipartite and an algorithm running in O(n 4 3 ) time to find the chaotic numbers of tripartite graphs where n 3 is the number of vertices in the biggest partite set.…”