Use of hidden network structure in the set partitioning problem

  • Agha Iqbal Ali
  • Hyun-Soo Han
  • Jeffery L. Kennington
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 920)


This paper demonstrates the use of hidden network structure for the solution of set partitioning problems. By finding a hidden network row submatrix, the set partitioning problem is transformed to a network with side constraints. Flow conditions on the revealed pure network are then used in a procedure for effecting variable reduction for the set partitioning. By finding a hidden network column submatrix the set partitioning problem is transformed to a network with side columns. The resulting formulation is used in finding a feasible solution for the set partitioning problem quickly. Computational experience is included.


Set Partitioning Integer Programming Networks Submatrix Transformation 


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  1. [1]
    Ali, A. I., H. Han. 1994a. “Computational Implementation of Fujishige's Graph Realizability Algorithm,” Working Paper, School of Management, University of Massachusetts at Amherst.Google Scholar
  2. [2]
    Ali, A. I., H. Han. 1994b. “Extracting Hidden Network Submatrices of a (0,1) Matrix Using a PQ-graph,” Working Paper, School of Management, University of Massachusetts at Amherst.Google Scholar
  3. [3]
    Ali, A. I., H. Thiagarajan. 1989. “A Network Relaxtion Based Enumeration Algorithm for Set Partioning,” European Journal of Operational Research, 38, 76–89.Google Scholar
  4. [4]
    Avis, A. 1980. “A Note on Some Computationally Difficult Set Covering problems,” Mathematical Programming, 18, 135–143.Google Scholar
  5. [5]
    Balas, E. 1980. “Cutting Planes from Conditional Bounds: A New Approach to Set Covering,” Mathematical Programming Study, 12, 19–36.Google Scholar
  6. [6]
    Balas, E., M. W. Padberg. 1979. Set Partioning — A Survey. In Combinatorial Optimization, Christofides, N., Mingozzi, P., Toth, P. and Sandi, C. (eds.). John Wiley, Chichester, England.Google Scholar
  7. [7]
    Bartholdi, J. 1982. “A Good Submatrix is Hard to Find,” Operations Research Letters 1, 190–193.Google Scholar
  8. [8]
    Bixby, R. E., R. Fourer. 1988. “Finding Embedded Network Rows in Linear Programs, I. Extraction Heuristics,” Management Science, 34, 342–376.Google Scholar
  9. [9]
    Brown, G., D. Thomen. 1980. “Automatic Identification of Generalized Upper Bounds in Large-scale Optimization Models,” Management Science, 26, 1166–1184.Google Scholar
  10. [10]
    Brown, G., R. McBride, R. Wood. 1985. “Extracting Embedded Generalized Networks from Linear Programming Problems,” Mathematical Programming, 32, 11–31.Google Scholar
  11. [11]
    Brown, G., W. Wright. 1984. “Automatic Identification of Embedded Network Rows in Large-scale Optimization Models,” Mathematical Programming, 29, 41–46.Google Scholar
  12. [12]
    Chan, T. J., C. A. Yano. 1992. “A Multiplier Adjustment Approach for the Set Partitioning Problem,” Operations Research, 40, Supp. 1, S40–S47.Google Scholar
  13. [13]
    Etcheberry, J. 1977. “The Set-Covering Problem: A New Implicit Enumeration Algorithm,” Operations Research, 25, 760–772.Google Scholar
  14. [14]
    Fisher, M. L., P. Kedia. 1990. “Optimal Solution of Set Covering/Partioning Problems Using Dual Heuristics,” Management Science, 36, 674–688.Google Scholar
  15. [15]
    Fujishige, S. 1980. “An Efficient PQ-Graph Algorithm for Solving the Graph Realization Problem,” Journal of Computer and System Sciences, 21, 63–86.Google Scholar
  16. [16]
    Graves, G., R. McBride, I. Gershkoff, D. Anderson, D. Mahidhara. 1993. “Flight Crew Scheduling,” Management Science, 39, 736–745.Google Scholar
  17. [17]
    Gunawardane, G., S. Hoff, L. Schrage. 1981. “Identification of Special Structure Constraints in Linear Programs,” Mathematical Programming, 21, 90–97.Google Scholar
  18. [18]
    Han, Hyun-Soo, “Algorithms to Extract Hidden Networks and Applications to Set Partitioning Problems,” unpublished dissertation, University of Massahusetts at Amherst, 1994.Google Scholar
  19. [19]
    Hoffman, K. L., M. Padberg. 1993. “Solving Airline Crew-Scheduling Problems by Branch and Cut,” Management Science, 39, 657–682.Google Scholar
  20. [20]
    Marsten, R. E. 1974. “An Algorithm for Large Set Partitioning Problems,” Management Science, 20, 191–209.Google Scholar
  21. [21]
    Nemhauser, G. L., G. L. Weber. 1979. “Optimal Set Partionings and Lagrangean Duality,” Naval Research Logistics Quartely, 26, 553–563.Google Scholar
  22. [22]
    Padberg, M. W. 1973. “On the Facial Structure of Set Packing Polyhedra,” Mathematical Programming, 5, 199–215.Google Scholar
  23. [23]
    Shepardson, F., R. E. Marsten. 1980. “A Lagrangean Relaxation Algorithm for the Two Duty Period Scheduling Problem,” Management Science, 26, 274–281.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Agha Iqbal Ali
    • 1
  • Hyun-Soo Han
    • 1
  • Jeffery L. Kennington
    • 2
  1. 1.School of ManagementUniversity of Massachusetts at AmherstUK
  2. 2.Southern Methodist UniversityUSA

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