Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-10T05:52:56.284Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the Chvátal-rankof clique family inequalities

Published online by Cambridge University Press:  21 August 2007

Arnaud Pêcher  and
Annegret K. Wagler
Arnaud Pêcher
Affiliation:
Université de Bordeaux (LaBRI, INRIA), 351 cours de la Libération, 33405 Talence, France; [email protected]
Annegret K. Wagler
Affiliation:
Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany; [email protected]
Get access

Abstract


Clique family inequalities a∑v∈W xv + (a - 1)∈v∈W , xv ≤ aδ form an intriguing class of valid inequalities for the stable set polytopes of all graphs. We prove firstly that their Chvátal-rank is at most a, which provides an alternative proof for the validity of clique family inequalities, involving only standard rounding arguments.Secondly, we strengthen the upper bound further and discuss consequences regarding the Chvátal-rank of subclasses of claw-free graphs.


Keywords

Stable set polytopeChvátal-rank
Type
Research Article
Information
RAIRO - Operations Research , Volume 41 , Issue 3: Journées Polyèdres et Optimisation Combinatoire , July 2007 , pp. 289 - 294
Copyright
© EDP Sciences, ROADEF, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A. Ben Rebea, Étude des stables dans les graphes quasi-adjoints. Ph.D. thesis, Univ. Grenoble (1981).
Cook, W., Kannan, R. and Schrijver, A., Chvátal closures for mixed integer programming problems. Math. Program. 47 (1990) 155–174. CrossRef
M. Chudnovsky and P. Seymour, Claw-free graphs VI. The structure of quasi-line graphs. manuscript (2004).
Chvátal, V., Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Math. 4 (1973) 305–337. CrossRef
V. Chvátal, On certain polytopes associated with graphs. J. Comb. Theory (B) 18 (1975) 138–154.
V. Chvátal, W. Cook and M. Hartmann, On cutting-plane proofs in combinatorial optimization. Linear Algebra Appl. 114/115 (1989) 455–499.
Eisenbrand, F., Oriolo, G., Stauffer, G. and Ventura, P., Circular one matrices and the stable set polytope of quasi-line graphs. Lect. Notes Comput. Sci. 3509 (2005) 291–305. CrossRef
R. Giles and L.E. Trotter Jr., On stable set polyhedra for K 1,3-free graphs. J. Comb. Theory B 31 (1981) 313–326.
Gomory, R.E., Outline of an algorithm for integer solutions to linear programs. Bull. Amer. Math. Soc. 64 (1958) 27–278. CrossRef
Liebling, T.M., Oriolo, G., Spille, B., and Stauffer, G., On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs. Math. Methods Oper. Res. 59 (2004) 25–35 CrossRef
Oriolo, G., On the Stable Set Polytope for Quasi-Line Graphs, Special issue on stability problems. Discrete Appl. Math. 132 (2003) 185–201. CrossRef
A. Pêcher and A. Wagler, Almost all webs are not rank-perfect. Math. Program. B 105 (2006) 311–328.
G. Stauffer, On the Stable Set Polytope of Claw-free Graphs. Ph.D. thesis, EPF Lausanne (2005).