Soft Computing

, Volume 23, Issue 9, pp 3013–3027 | Cite as

Computational study of separation algorithms for clique inequalities

  • Francesca Marzi
  • Fabrizio Rossi
  • Stefano SmriglioEmail author


Clique inequalities appear in linear descriptions of many combinatorial optimisation problems. In general, they form an exponential family and, in addition, the associated separation problem is strongly NP-hard, being equivalent to a maximum weight clique problem. Therefore, most of the known (both exact and heuristic) separation procedures follow the decomposition scheme of a maximum clique algorithm. We introduce a new heuristic, aimed at constructing a collection of (violated) clique inequalities covering all the edges of the underlying graph. We present an extensive computational experience showing that this closely approximates the results of an exact separation oracle while being faster than standard heuristics.


Clique inequalities Separation problem Cutting plane algorithm 



This study was funded by Italian Ministry of Education and Research, National Research Program PRIN 2015, Grant No. 20153TXRX9.

Compliance with ethical standards

Conflict of interest

All authors declare that they have no conflict of interest

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Atamtürk A, Nemhauser GL, Savelsbergh MWP (2000) Conflict graphs in solving integer programming problems. Eur J Oper Res 121(1):40–55MathSciNetCrossRefzbMATHGoogle Scholar
  2. Avella P, Boccia M, Mannino C, Vasilyev I (2017) Time-indexed formulations for the runway scheduling problem. Transp Sci 51(4):1196–1209CrossRefGoogle Scholar
  3. Balas E, Padberg MW (1976) Set partitioning: a survey. Management sciences research report, Graduate School of Industrial Administration, Carnegie-Mellon UniversityGoogle Scholar
  4. Borndörfer R (1998) Aspects of set packing, partitioning, and covering. Ph. D. thesis, TU BerlinGoogle Scholar
  5. Borndörfer R, Kormos Z (1997) An algorithm for maximum cliques. Unpublished working paper, Konrad-Zuse-Zentrum für InformationstechnikGoogle Scholar
  6. Borndörfer R, Weismantel R (2000) Set packing relaxations of some inte]ger programs. Math Program 88(3):425–450MathSciNetCrossRefzbMATHGoogle Scholar
  7. Caprara A, Salazar González JJ (1999) Separating lifted odd-hole inequalities to solve the index selection problem. Discret Appl Math 92(2):111–134MathSciNetCrossRefzbMATHGoogle Scholar
  8. Carraghan R, Pardalos PM (1990) An exact algorithm for the maximum clique problem. Oper Res Lett 9(6):375–382CrossRefzbMATHGoogle Scholar
  9. Corrêa RC, Delle Donne D, Koch I, Marenco J (2017) General cut-generating procedures for the stable set polytope. Discret Appl Math 245:28–41MathSciNetCrossRefzbMATHGoogle Scholar
  10. de Givry S, Katsirelos G (2017) Clique cuts in weighted constraint satisfaction. In: Beck JC (ed) Principles and practice of constraint programming. Springer International Publishing, Cham, pp 97–113CrossRefGoogle Scholar
  11. Escudero LF, Landete M, Marín A (2009) A branch-and-cut algorithm for the winner determination problem. Decis Support Syst 46(3):649–659CrossRefGoogle Scholar
  12. Groiez M, Desaulniers G, Marcotte O (2014) Valid inequalities and separation algorithms for the set partitioning problem. INFOR: Inf Syst Oper Res 52(4):185–196MathSciNetGoogle Scholar
  13. Grötschel M, Lovász L, Schrijver A (1988) Geometric algorithms and combinatorial optimization. Springer, BerlinCrossRefzbMATHGoogle Scholar
  14. Grötschel M, Lovász L, Schrijver A (1981) The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2):169–197MathSciNetCrossRefzbMATHGoogle Scholar
  15. Håstad J (1999) Clique is hard to approximate within \(n^{(1-\epsilon )}\). Acta Math 182(1):105–142MathSciNetCrossRefzbMATHGoogle Scholar
  16. Hoffman K, Padberg M (1993) Solving airline crew scheduling problems by branch-and-cut. Manag Sci 39(6):657–682CrossRefzbMATHGoogle Scholar
  17. Intel distribution for Python (2018)
  18. Johnson DJ, Trick MA (eds) (1996) Cliques, coloring, and satisfiability: second DIMACS implementation challenge, Workshop, October 11–13, 1993. American Mathematical Society, Boston, MAGoogle Scholar
  19. Letchford AN, Rossi F, Smriglio S (2018) The stable set problem: clique and nodal inequalities revisited. Optimization Online,
  20. Nemhauser GL, Sigismondi G (1992) A strong cutting plane/branch-and-bound algorithm for node packing. J Oper Res Soc 43(5):443–457CrossRefzbMATHGoogle Scholar
  21. Niskanen S, Östergård PRJ (2002) Routines for clique searching.
  22. Padberg MW (1973) On the facial structure of set packing polyhedra. Math Program 5(1):199–215MathSciNetCrossRefzbMATHGoogle Scholar
  23. Pardalos P, Rappe J, Resende MGC (1998) An exact parallel algorithm for the maximum clique problem. In: De Leone R et al (eds) High performance algorithms and software in nonlinear optimization. Springer, Boston, pp 279–300CrossRefGoogle Scholar
  24. Rossi F, Smriglio S (2001) A set packing model for the ground holding problem in congested networks. Eur J Oper Res 131(2):400–416MathSciNetCrossRefzbMATHGoogle Scholar
  25. Spoorendonk S, Desaulniers G (2010) Clique inequalities applied to the vehicle routing problem with time windows. INFOR J 48(1):53–67MathSciNetGoogle Scholar
  26. SCIP Optimization Suite (2018).
  27. Waterer H, Johnson EL, Nobili P, Savelsbergh MWP (2002) The relation of time indexed formulations of single machine scheduling problems to the node packing problem. Math Program 93(3):477–494MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.DISIM, University of L’AquilaL’AquilaItaly

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