Skip to main content
SpringerLink
Log in

Lovász-Schrijver PSD-Operator on Claw-Free Graphs

  • Conference paper
  • First Online:
Combinatorial Optimization (ISCO 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9849))

Included in the following conference series:

Abstract

The subject of this work is the study of the Lovász-Schrijver PSD-operator \(LS_+\) applied to the edge relaxation \(\mathrm{ESTAB}(G)\) of the stable set polytope \(\mathrm{STAB}(G)\) of a graph G. We are interested in the problem of characterizing the graphs G for which \(\mathrm{STAB}(G)\) is achieved in one iteration of the \(LS_+\)-operator, called \(LS_+\)-perfect graphs, and to find a polyhedral relaxation of \(\mathrm{STAB}(G)\) that coincides with \(LS_+(\mathrm{ESTAB}(G))\) and \(\mathrm{STAB}(G)\) if and only if G is \(LS_+\)-perfect. An according conjecture has been recently formulated (\(LS_+\)-Perfect Graph Conjecture); here we verify it for the well-studied class of claw-free graphs.

This work was supported by an ECOS-MINCyT cooperation (A12E01), a MATH-AmSud cooperation (PACK-COVER), PID-CONICET 0277, PICT-ANPCyT 0586.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Bianchi, S., Escalante, M., Nasini, G., Tunçel, L.: Near-perfect graphs with polyhedral \(N_+(G)\). Electron. Notes Discrete Math. 37, 393–398 (2011)

    Article  MATH  Google Scholar 

  2. Bianchi, S., Escalante, M., Nasini, G., Tunçel, L.: Lovász-Schrijver PSD-operator and a superclass of near-perfect graphs. Electron. Notes Discrete Math. 44, 339–344 (2013)

    Article  Google Scholar 

  3. Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. 164, 51–229 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chudnovsky, M., Seymour, P.: The structure of claw-free graph (unpliblished manuscript) (2004)

    Google Scholar 

  5. Chvátal, V.: On certain polytopes associated with graphs. J. Comb. Theory (B) 18, 138–154 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coulonges, S., Pêcher, A., Wagler, A.: Characterizing and bounding the imperfection ratio for some classes of graphs. Math. Program. A 118, 37–46 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Edmonds, J.R.: Maximum matching and a polyhedron with (0,1) vertices. J. Res. Nat. Bur. Stand. 69B, 125–130 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  8. Edmonds, J.R., Pulleyblank, W.R.: Facets of I-matching polyhedra. In: Berge, C., Chuadhuri, D.R. (eds.) Hypergraph Seminar. LNM, pp. 214–242. Springer, Heidelberg (1974)

    Google Scholar 

  9. Eisenbrand, F., Oriolo, G., Stauffer, G., Ventura, P.: The stable set polytope of quasi-line graphs. Combinatorica 28, 45–67 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Escalante, M., Montelar, M.S., Nasini, G.: Minimal \(N_+\)-rank graphs: progress on Lipták and Tunçel’s conjecture. Oper. Res. Lett. 34, 639–646 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Escalante, M., Nasini, G.: Lovász and Schrijver \(N_+\)-relaxation on web graphs. In: Fouilhoux, P., Gouveia, L.E.N., Mahjoub, A.R., Paschos, V.T. (eds.) ISCO 2014. LNCS, vol. 8596, pp. 221–229. Springer, Heidelberg (2014)

    Google Scholar 

  12. Escalante, M., Nasini, G., Wagler, A.: Characterizing \(N_+\)-perfect line graphs. Int. Trans. Oper. Res. (2016, to appear)

    Google Scholar 

  13. Galluccio, A., Gentile, C., Ventura, P.: Gear composition and the stable set polytope. Oper. Res. Lett. 36, 419–423 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Galluccio, A., Gentile, C., Ventura, P.: The stable set polytope of claw-free graphs with stability number at least four. I. Fuzzy antihat graphs are W-perfect. J. Comb. Theory Ser. B 107, 92–122 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Galluccio, A., Gentile, C., Ventura, P.: The stable set polytope of claw-free graphs with stability number at least four. II. Striped graphs are G-perfect. J. Comb. Theory Ser. B 108, 1–28 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Giles, R., Trotter, L.E.: On stable set polyhedra for K1,3 -free graphs. J. Comb. Theory 31, 313–326 (1981)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, New York (1988)

    Book  MATH  Google Scholar 

  19. Liebling, T.M., Oriolo, G., Spille, B., Stauffer, G.: On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs. Math. Methods Oper. Res. 59(1), 25–35 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lipták, L., Tunçel, L.: Stable set problem and the lift-and-project ranks of graphs. Math. Program. A 98, 319–353 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25, 1–7 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1, 166–190 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. Minty, G.: On maximal independent sets of vertices in claw-free graphs. J. Comb. Theory 28, 284–304 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Oriolo, G., Pietropaoli, U., Stauffer, G.: A new algorithm for the maximum weighted stable set problem in claw-free graphs. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 77–96. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  25. Pêcher, A., Wagler, A.: Almost all webs are not rank-perfect. Math. Program. B 105, 311–328 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pêcher, A., Wagler, A.: On facets of stable set polytopes of claw-free graphs with stability number three. Discrete Math. 310, 493–498 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pietropaoli, U., Wagler, A.: Some results towards the description of the stable set polytope of claw-free graphs. In: Proceedings of ALIO/EURO Workshop on Applied Combinatorial Optimization, Buenos Aires (2008)

    Google Scholar 

  28. Sbihi, N.: Algorithme de recherche d’un stable de cardinalité maximum dans un graphe sans étoile. Discrete Math. 29, 53–76 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shepherd, F.B.: Near-perfect matrices. Math. Program. 64, 295–323 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. Shepherd, F.B.: Applying Lehman’s theorem to packing problems. Math. Program. 71, 353–367 (1995)

    MathSciNet  MATH  Google Scholar 

  31. Stauffer, G.: On the stable set polytope of claw-free graphs. Ph.D. thesis, Swiss Institute of Technology in Lausanne (2005)

    Google Scholar 

  32. Wagler, A.: Critical edges in perfect graphs. Ph.D. thesis, TU Berlin and Cuvillier Verlag Göttingen (2000)

    Google Scholar 

  33. Wagler, A.: Antiwebs are rank-perfect. 4OR 2, 149–152 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. FCEIA, Universidad Nacional de Rosario, Rosario, Argentina

    Silvia Bianchi, Mariana Escalante & Graciela Nasini

  2. LIMOS (UMR 6158 CNRS), University Blaise Pascal, Clermont-Ferrand, France

    Annegret Wagler

Authors
  1. Silvia Bianchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mariana Escalante
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Graciela Nasini
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Annegret Wagler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Annegret Wagler .

Editor information

Editors and Affiliations

  1. University of Salerno , Fisciano, Italy

    Raffaele Cerulli

  2. Kyoto University , Kyoto, Japan

    Satoru Fujishige

  3. LAMSADE, Université Paris-Dauphine , Paris , France

    A. Ridha Mahjoub

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Bianchi, S., Escalante, M., Nasini, G., Wagler, A. (2016). Lovász-Schrijver PSD-Operator on Claw-Free Graphs. In: Cerulli, R., Fujishige, S., Mahjoub, A. (eds) Combinatorial Optimization. ISCO 2016. Lecture Notes in Computer Science(), vol 9849. Springer, Cham. https://doi.org/10.1007/978-3-319-45587-7_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-45587-7_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45586-0

  • Online ISBN: 978-3-319-45587-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation