EURO Journal on Computational Optimization

, Volume 6, Issue 2, pp 185–206 | Cite as

Improved integral simplex using decomposition for the set partitioning problem

  • Abdelouahab Zaghrouti
  • Issmail El HallaouiEmail author
  • François Soumis
Original Paper


Integral simplex using decomposition (ISUD) is a method that efficiently solves set partitioning problems. It is an iterative method that starts from a known integer solution and moves through a sequence of integer solutions, decreasing the cost at each iteration. At each iteration, the method decomposes the original problem into a reduced problem (RP) and a complementary problem (CP). Given an integer solution to RP (that is also solution to the original problem), CP finds a descent direction having the minimum ratio between its cost and the number of its positive variables. We loop on until an optimal or near-optimal solution to the original problem is reached. In this paper, we introduce a modified model for CP. The new model finds a descent direction that minimizes the ratio between the cost of the direction and an overestimation of the number of variables taking one in the next solution. The new CP presents higher chances of finding improved integer solutions without branching. We present results for the same large instances (with up to 570,000 columns) as the ones previously used to test ISUD. For all the instances, optimality is always reached with a speedup factor of at least five.


Primal algorithms Set partitioning problem Integral simplex 

Mathematics Subject Classification

90C99 90C27 90C10 


  1. Balas E, Padberg MW (1972) On the set-covering problem. Oper Res 20:1152–1161CrossRefGoogle Scholar
  2. Balas E, Padberg MW (1975) On the set-covering problem: II. An algorithm for set partitioning. Oper Res 23:74–90CrossRefGoogle Scholar
  3. El Hallaoui I, Metrane A, Desaulniers G, Soumis F (2011) An improved primal simplex algorithm for degenerate linear programs. INFORMS J Comput 23(4):569–577CrossRefGoogle Scholar
  4. Haus U, Köppe M, Weismantel R (2001) The integral basis method for integer programming. Math Methods Oper Res 53(3):353–361CrossRefGoogle Scholar
  5. Rönnberg E, Larsson T (2009) Column generation in the integral simplex method. Eur J Oper Res 192(1):333–342CrossRefGoogle Scholar
  6. Rosat S, El Hallaoui I, Soumis F, Lodi A (2014) Integral simplex using decomposition with primal cuts. In: Experimental algorithms. Springer International Publishing, pp 22–33Google Scholar
  7. Rosat S, El Halaoui I, Soumis F, Chakour D (2017) Influence of the normalization constraint on the integral simplex using decomposition. Discrete Appl Math 217(1):53–70Google Scholar
  8. Saxena A (2003) Set-partitioning via integral simplex method. Doctoral dissertation, Ph. D. thesis, Carnegie Mellon University, PittsburghGoogle Scholar
  9. Thompson GL (2002) An integral simplex algorithm for solving combinatorial optimization problems. Comput Optim Appl 22:351–367CrossRefGoogle Scholar
  10. Zaghrouti A, El Hallaoui I, Soumis F (2013) Improving ILP solutions by zooming around an improving direction. Les Cahiers du GERAD G-2013-107Google Scholar
  11. Zaghrouti A, Soumis F, El Hallaoui I (2014) Integral simplex using decomposition for the set partitioning problem. Oper Res 62(2):435–449CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and EURO - The Association of European Operational Research Societies 2018

Authors and Affiliations

  • Abdelouahab Zaghrouti
    • 1
  • Issmail El Hallaoui
    • 1
    Email author
  • François Soumis
    • 1
  1. 1.Département de Mathématiques et de Génie IndustrielPolytechnique Montreal and GERADMontrealCanada

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