Journal of the Korea Society of Computer and Information (한국컴퓨터정보학회논문지)

Korean Society of Computer Information (한국컴퓨터정보학회)
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The Grid Type Quadratic Assignment Problem Algorithm

그리드형 2차 할당문제 알고리즘

  • Lee, Sang-Un (Dept. of Multimedia Eng., Gangneung-Wonju National University)
  • 이상운 (강릉원주대학교 멀티미디어공학과)
  • Received : 2014.02.18
  • Accepted : 2014.03.25
  • Published : 2014.04.30
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Abstract

TThis paper suggests an heuristic polynomial time algorithm to solve the optimal solution for QAP (quadratic assignment problem). While Hungarian algorithm is most commonly used for a linear assignment, there is no polynomial time algorithm for the QAP. The proposed algorithm derives a grid type layout among unit distances of a distance matrix. And, we apply max-flow/min-distance approach to assign this grid type layout in such a descending order way that the largest flow is matched to the smallest unit distance from flow matrix. Evidences from implementation results of the proposed algorithm on various numerical grid type QAP examples show that a solution to the QAP could be obtained by a polynomial algorithm.

본 논문은 2차 할당 문제의 최적 해를 찾을 수 있는 휴리스틱 다항시간 알고리즘을 제안하였다. 일반적으로 선형 할당 문제의 최적 해는 헝가리안 알고리즘으로 구한다. 그러나 2차 할당문제의 최적 해를 찾는 알고리즘은 제안되지 않고 있다. 제안된 알고리즘은 거리행렬의 단위거리로부터 그리드 형태의 배치를 찾아내고, 이 형태에 맞도록 흐름량 행렬의 최대 값부터 내림차순으로 최대흐름을 최소 거리에 배치하는 최대흐름/최소거리 기법을 적용하였다. 제안된 알고리즘을 다양한 그리드형 2차 할당문제에 적용한 결과 2차 할당문제의 해를 다항시간에 구할 수 있는 알고리즘이 존재할 가능성을 보였다.

Keywords

References

  1. B. Rainer, M. Dell'Amico, and S. Martello, "Assignment Problems," SIAM, 2009.
  2. D. N. Kumar, "Optimization Methods," http://www.nptel.iitm.ac.in/Courses/Webcourse-contents/IISc-BANG/OPTIMIZATIONMETHODS/pdf/Module_4/M4L3_LN.pdf, IISc, Bangalore, 2008.
  3. E. M. Loiola, N. M. M. Abreu, P. O. Boaventura-Netto, P. Hahn, and T. Querido, "A Survey of the Quadratic Assignment Problem," European Journal of Operational Research, Vol. 176, No. 2, pp. 657-690, Jan. 2007. https://doi.org/10.1016/j.ejor.2005.09.032
  4. P. Ji, Y. Wu, and H. Liu, "A Solution Method for the Quadratic Assignment Problem (QAP)," The 6th International Symposium on Operations Research and Its Applications (ISORA'06), Xinjiang, China, pp. 106-117, Aug. 2006.
  5. L. Steinberg, "The Backboard Wiring Problem: A Placement Algorithm," SIAM Review, Vol. 3, No. 1, pp. 37-50, Jan. 1961. https://doi.org/10.1137/1003003
  6. A. N. Elshafei, "Hospital Layout as a Quadratic Assignment Problem," Journal of the Operational Research Society, Vol. 28, pp. 167-179, Apr. 1977. https://doi.org/10.1057/jors.1977.29
  7. H. W. Kuhn, "The Hungarian Method for the Assignment Problem," Naval Research Logistics Quarterly, Vol. 2, No. 1-2, pp. 83-97, Mar. 1955. https://doi.org/10.1002/nav.3800020109
  8. R. E. Burkard, S. E. Karisch, and F. Rendl, "QAPLIB-A Quadratic Assignment Problem Library," http://www.seas.upenn.edu/ qaplib/, Journal of Global Optimization, Vol. 10, No. 4, pp. 391-403, Jun. 1997. https://doi.org/10.1023/A:1008293323270
  9. P. M. Pardolas and J. V. Crouse, "A Parallel Algorithm for the Quadratic Assignment Problem," Proceedings of the ACM/IEEE Conference on Supercomputing, pp. 351-360, Nov. 1989.
  10. S. U. Lee, "A Reverse-delete Algorithm for Assignment Problem," Journal of Korean Institute of Information Technology, Vol. 10, No. 8, pp. 117-126, Aug. 2012.
  11. S. U. Lee, "The Optimal Algorithm for Assignment Problem," Journal of The Korea Society of Computer and Information, Vol. 17, No. 9, pp. 139-147, Sep. 2012. https://doi.org/10.9708/jksci/2012.17.9.139
  12. A. A. Assad and W. Xu, "On Lower Bounds for a Class of Quadratic {0,1} Programs," Operations Research Letters, Vol. 4, No. 4, pp. 175-180, Dec. 1985. https://doi.org/10.1016/0167-6377(85)90025-2