Skip to main content
SpringerLink
Log in

Computational study of separation algorithms for clique inequalities

  • Focus
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Atamtürk A, Nemhauser GL, Savelsbergh MWP (2000) Conflict graphs in solving integer programming problems. Eur J Oper Res 121(1):40–55

    Article  MathSciNet  MATH  Google Scholar 

  • Avella P, Boccia M, Mannino C, Vasilyev I (2017) Time-indexed formulations for the runway scheduling problem. Transp Sci 51(4):1196–1209

    Article  Google Scholar 

  • Balas E, Padberg MW (1976) Set partitioning: a survey. Management sciences research report, Graduate School of Industrial Administration, Carnegie-Mellon University

  • Borndörfer R (1998) Aspects of set packing, partitioning, and covering. Ph. D. thesis, TU Berlin

  • Borndörfer R, Kormos Z (1997) An algorithm for maximum cliques. Unpublished working paper, Konrad-Zuse-Zentrum für Informationstechnik

  • Borndörfer R, Weismantel R (2000) Set packing relaxations of some inte]ger programs. Math Program 88(3):425–450

    Article  MathSciNet  MATH  Google Scholar 

  • Caprara A, Salazar González JJ (1999) Separating lifted odd-hole inequalities to solve the index selection problem. Discret Appl Math 92(2):111–134

    Article  MathSciNet  MATH  Google Scholar 

  • Carraghan R, Pardalos PM (1990) An exact algorithm for the maximum clique problem. Oper Res Lett 9(6):375–382

    Article  MATH  Google Scholar 

  • Corrêa RC, Delle Donne D, Koch I, Marenco J (2017) General cut-generating procedures for the stable set polytope. Discret Appl Math 245:28–41

    Article  MathSciNet  MATH  Google Scholar 

  • 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 97–113

    Chapter  Google Scholar 

  • Escudero LF, Landete M, Marín A (2009) A branch-and-cut algorithm for the winner determination problem. Decis Support Syst 46(3):649–659

    Article  Google Scholar 

  • Groiez M, Desaulniers G, Marcotte O (2014) Valid inequalities and separation algorithms for the set partitioning problem. INFOR: Inf Syst Oper Res 52(4):185–196

    MathSciNet  Google Scholar 

  • Grötschel M, Lovász L, Schrijver A (1988) Geometric algorithms and combinatorial optimization. Springer, Berlin

    Book  MATH  Google Scholar 

  • Grötschel M, Lovász L, Schrijver A (1981) The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2):169–197

    Article  MathSciNet  MATH  Google Scholar 

  • Håstad J (1999) Clique is hard to approximate within \(n^{(1-\epsilon )}\). Acta Math 182(1):105–142

    Article  MathSciNet  MATH  Google Scholar 

  • Hoffman K, Padberg M (1993) Solving airline crew scheduling problems by branch-and-cut. Manag Sci 39(6):657–682

    Article  MATH  Google Scholar 

  • Intel distribution for Python (2018) https://software.intel.com/en-us/distribution-for-python

  • Johnson DJ, Trick MA (eds) (1996) Cliques, coloring, and satisfiability: second DIMACS implementation challenge, Workshop, October 11–13, 1993. American Mathematical Society, Boston, MA

  • Letchford AN, Rossi F, Smriglio S (2018) The stable set problem: clique and nodal inequalities revisited. Optimization Online, http://www.optimization-online.org/DB_HTML/2018/05/6612.html

  • Nemhauser GL, Sigismondi G (1992) A strong cutting plane/branch-and-bound algorithm for node packing. J Oper Res Soc 43(5):443–457

    Article  MATH  Google Scholar 

  • Niskanen S, Östergård PRJ (2002) Routines for clique searching. http://users.aalto.fi/~pat/cliquer.html

  • Padberg MW (1973) On the facial structure of set packing polyhedra. Math Program 5(1):199–215

    Article  MathSciNet  MATH  Google Scholar 

  • 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 279–300

    Chapter  Google Scholar 

  • Rossi F, Smriglio S (2001) A set packing model for the ground holding problem in congested networks. Eur J Oper Res 131(2):400–416

    Article  MathSciNet  MATH  Google Scholar 

  • Spoorendonk S, Desaulniers G (2010) Clique inequalities applied to the vehicle routing problem with time windows. INFOR J 48(1):53–67

    MathSciNet  Google Scholar 

  • SCIP Optimization Suite (2018). http://scip.zib.de/

  • 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):477–494

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. DISIM, University of L’Aquila, Via Vetoio, Coppito, 67010, L’Aquila, AQ, Italy

    Francesca Marzi, Fabrizio Rossi & Stefano Smriglio

Authors
  1. Francesca Marzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabrizio Rossi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stefano Smriglio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefano Smriglio.

Ethics declarations

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.

Additional information

Communicated by P. Beraldi, M. Boccia, C. Sterle.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Marzi, F., Rossi, F. & Smriglio, S. Computational study of separation algorithms for clique inequalities. Soft Comput 23, 3013–3027 (2019). https://doi.org/10.1007/s00500-019-03769-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-019-03769-y

Keywords

Search

Navigation