Integrating combinatorial algorithms into a linear programming solver

  • Richárd Molnár-Szipai
  • Anita VargaEmail author
Original Paper


While there are numerous linear (and nonlinear) solvers, as well as specialized algorithms for combinatorial problems, they are rarely used together. We wrote a new module for the XPRESS optimizer that lets us call the objects and functions of the LEMON C++ library. We tested this module by comparing two versions of the Dual Ascent Procedure (DAP) (Adams and Johnson in DIMACS Ser Discrete Math Theor Comput Sci 16:43–75, 1994). It is an iterative algorithm that produces lower bounds for the quadratic assignment problem (QAP). The DAP is based on the level-1 RLT-relaxation (reformulation-linearization technique) of the QAP. This formulation is significantly larger than the original QAP, but if we construct its Lagrangian dual, we can decompose the new relaxation to smaller linear assignment problems for a fixed Lagrange multiplier. The algorithm determines a better Lagrange multiplier in every iteration. We implemented two versions of the DAP with the XPRESS Optimization Software. In the first model we solved the smaller linear assignment problems with the linear programming methods of XPRESS, while in the second model we solved them by using the graph algorithms of the LEMON library via the new module. Using the instances of QAPLIB, the new module produced significantly better running times, which suggests that this direction of research might yield further results in the future.


Combinatorial optimization XPRESS optimization software LEMON C++ library Quadratic assignment problem 



We would like to thank Dr. Zsolt Csizmadia, lead software engineer of FICO. Without his help this work would not have been possible.


  1. Adams WP, Guignard M, Hahn PM, Hightower WL (2007) A level-2 reformulation linearization technique bound for the quadratic assignment problem. Eur J Oper Res 180(3):983–996. CrossRefGoogle Scholar
  2. Adams WP, Johnson TA (1994) Improved linear programming-based lower bounds for the quadratic assignment problem. DIMACS Ser Discrete Math Theor Comput Sci 16:43–75CrossRefGoogle Scholar
  3. Burkard R, Derigs U (1980) Lecture notes in economics and mathematical systems. Lectures Notes in Economics and Mathematical Systems 184Google Scholar
  4. Burkard R, Karisch S, Rendl F (1997) QAPLIB: a quadratic assignment problem library. J Glob Optim 10:391–403CrossRefGoogle Scholar
  5. Burkard RE, Çela E, Pardalos PM, Pitsoulis LS (1998) The quadratic assignment problem. Springer, BerlinGoogle Scholar
  6. Chaovalitwongse W, Pardalos PM, Prokopyev OA (2004) A new linearization technique for multi-quadratic 0–1 programming problems. Oper Res Lett 32(6):517–522CrossRefGoogle Scholar
  7. Clausen J, Perregaard M (1997) Solving large quadratic assignment problems in parallel. Comput Optim App 8(2):111–127CrossRefGoogle Scholar
  8. Dezső B, Jüttner A, Kovács P (2011) Lemon-an open source c++ graph template library. Electron Notes Theor Comput Sci 264(5):23–45CrossRefGoogle Scholar
  9. Everett H III (1963) Generalized lagrange multiplier method for solving problems of optimum allocation of resources. Oper Res 11(3):399–417CrossRefGoogle Scholar
  10. Hahn PM, Krarup J (2001) A hospital facility layout problem finally solved. J Intell Manuf 12(5–6):487–496CrossRefGoogle Scholar
  11. Hahn PM, Zhu YR, Guignard M, Hightower WL, Saltzman MJ (2012) A level-3 reformulation-linearization technique-based bound for the quadratic assignment problem. INFORMS J Comput 24(2):202–209CrossRefGoogle Scholar
  12. Hillier FS, Connors MM (1966) Quadratic assignment problem algorithms and the location of indivisible facilities. Manag Sci 13(1):42–57CrossRefGoogle Scholar
  13. Koopmans TC, Beckmann M (1957) Assignment problems and the location of economic activities. Econom: J Econom Soc 25(1):53–76Google Scholar
  14. Krarup J, Pruzan PM (1978) Computer-aided layout design. In: Mathematical programming in use. Springer, Berlin, pp 75–94Google Scholar
  15. Lawler EL (1963) The quadratic assignment problem. Manag Sci 9(4):586–599CrossRefGoogle Scholar
  16. Niroomand S, Takács S, Vizvári B (2011) To lay out or not to lay out? Ann Oper Res 191(1):183–192CrossRefGoogle Scholar
  17. Nugent CE, Vollmann TE, Ruml J (1968) An experimental comparison of techniques for the assignment of facilities to locations. Oper Res 16(1):150–173CrossRefGoogle Scholar
  18. Optimization D (2004) Xpress-mosel: user guide. Englewood Cliffs, NJGoogle Scholar
  19. Sahni S, Gonzalez T (1976) P-complete approximation problems. J ACM 23(3):555–565CrossRefGoogle Scholar
  20. Taassori M, Niroomand S, Uysal S, Vizvari B, Hadi-Vencheh A (2017) Optimization approaches for core mapping on networks on chip. IETE J Res 1–12.
  21. Taassori M, Taassori M, Niroomand S, Vizvári B, Uysal S, Hadi-Vencheh A (2015) Opaic: an optimization technique to improve energy consumption and performance in application specific network on chips. Measurement 74:208–220CrossRefGoogle Scholar
  22. Vizvári B (1978) Lagrange multipliers in integer programming. Probl Control Inf Theory 7(5):393–406Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Budapest University of Technology and EconomicsBudapestHungary

Personalised recommendations