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

Author

Listed:
  • H. L. Fu
  • C. L. Shiue
  • X. Cheng
  • D. Z. Du
  • J. M. Kim

Abstract

Let α be a permutation of the vertex set V(G) of a connected graph G. Define the total relative displacement of α in G by be $$\delta _\alpha (G) = \mathop \Sigma \limits_{x,y \in V(G)} |d_G (x,y) - d_G (\alpha (x),\alpha (y))|,$$ where dG(x, y) is the length of the shortest path between x and y in G. Let π* (G) be the maximum value of δα (G) among all permutations of V(G). The permutation which realizes π* (G) is called a chaotic mapping of G. In this paper, we study the chaotic mappings of complete multipartite graphs. The problem is reduced to a quadratic integer programming problem. 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.

Suggested Citation

  • H. L. Fu & C. L. Shiue & X. Cheng & D. Z. Du & J. M. Kim, 2001. "Quadratic Integer Programming with Application to the Chaotic Mappings of Complete Multipartite Graphs," Journal of Optimization Theory and Applications, Springer, vol. 110(3), pages 545-556, September.
    • Handle: RePEc:spr:joptap:v:110:y:2001:i:3:d:10.1023_a:1017584227417
    DOI: 10.1023/A:1017584227417
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    Citations

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    Cited by:

    1. Alain Billionnet & Sourour Elloumi & Amélie Lambert, 2013. "An efficient compact quadratic convex reformulation for general integer quadratic programs," Computational Optimization and Applications, Springer, vol. 54(1), pages 141-162, January.
    2. Dein Wong & Jiahui Yin & Jiao Wang, 2023. "Near automorphisms of complement or square of a path," Journal of Combinatorial Optimization, Springer, vol. 45(2), pages 1-10, March.
    3. Alain Billionnet & Sourour Elloumi & Amélie Lambert, 2014. "A Branch and Bound algorithm for general mixed-integer quadratic programs based on quadratic convex relaxation," Journal of Combinatorial Optimization, Springer, vol. 28(2), pages 376-399, August.
    4. N. P. Chiang, 2006. "Chaotic Numbers of Complete Bipartite Graphs and Tripartite Graphs," Journal of Optimization Theory and Applications, Springer, vol. 131(3), pages 485-491, December.

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