Skip to main content
Log in

A simplex like approach based on star sets for recognizing convex-\(QP\) adverse graphs

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

A graph \(G\) with convex-\(QP\) stability number (or simply a convex-\(QP\) graph) is a graph for which the stability number is equal to the optimal value of a convex quadratic program, say \(P(G)\). There are polynomial-time procedures to recognize convex-\(QP\) graphs, except when the graph \(G\) is adverse or contains an adverse subgraph (that is, a non complete graph, without isolated vertices, such that the least eigenvalue of its adjacency matrix and the optimal value of \(P(G)\) are both integer and none of them changes when the neighborhood of any vertex of \(G\) is deleted). In this paper, from a characterization of convex-\(QP\) graphs based on star sets associated to the least eigenvalue of its adjacency matrix, a simplex-like algorithm for the recognition of convex-\(QP\) adverse graphs is introduced.

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

Access this article

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

  • Abello J, Butenko S, Pardalos PM, Resende MGC (2001) Finding independent sets in a graph using continuous multivariable polynomial formulations. J Global Optim 21:111–137

    Article  MathSciNet  MATH  Google Scholar 

  • Bomze IM (1998) On standard quadratic optimization problems. J Global Optim 13:369–387

    Article  MathSciNet  MATH  Google Scholar 

  • Cardoso DM (2001) Convex quadratic programming approach to the maximum matching problem. J Global Optim 21:91–106

    Article  MathSciNet  Google Scholar 

  • Cvetković D, Rowlinson P, Simić S (2010) An introduction to the theory of graph spectra. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Doob M (1982) A suprising property of the least eigenvalue of a graph. Linear Algebra and Its Appl 46:1–7

    Article  MathSciNet  MATH  Google Scholar 

  • Kochenberger GA, Glover F, Alidaee B, Rego C (2013) An unconstrained quadratic binary programming Approach ao the vertex coloring problem. In: Pardalos P, Du, D, Graham RL (Eds) Handbook of combinatorial optimization, 2nd edn. Vol I. Springer, New York, p 533–558

  • Luz CJ (1995) An upper bound on the independence number of a graph computable in polynomial time. Operat Res Lett 18:139–145

    Article  MathSciNet  MATH  Google Scholar 

  • Luz CJ (2005) Improving an upper bound on the stability number of a graph. J Global Optim 31:61–84

    Article  MathSciNet  MATH  Google Scholar 

  • Luz CJ, Cardoso DM (1998) A generalization of the Hoffman-Lovász upper bound on the independence number of a regular graph. Ann Operat Res 81:307–319

    Article  MathSciNet  MATH  Google Scholar 

  • Motzkin TS, Straus EG (1965) Maxima for graphs and a new proof of a theorem of Turán. Can J Math 17:533–540

    Article  MathSciNet  MATH  Google Scholar 

  • Papadimitriou CH, Steiglitz K (1998) Combinatorial optimization: algorithms and complexity. Dover Publications, New York

    MATH  Google Scholar 

Download references

Acknowledgments

The authors thank the two referees for their helpful comments and suggestions that improved the paper. The research of both authors is partially supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT-Fundação para a Ciência e a Tecnologia”), within project PEst-OE/MAT/UI4106/2014.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Carlos J. Luz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cardoso, D.M., Luz, C.J. A simplex like approach based on star sets for recognizing convex-\(QP\) adverse graphs. J Comb Optim 31, 311–326 (2016). https://doi.org/10.1007/s10878-014-9745-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-014-9745-x

Keywords

Navigation