A note on the Chvátal-rank of clique family inequalities
RAIRO - Operations Research - Recherche Opérationnelle, Tome 41 (2007) no. 3, pp. 289-294.

Clique family inequalities a∑ v∈W x v +(a-1)∑ v∈W , x v ≤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.

DOI : 10.1051/ro:2007022
Classification : 05C69, 90C10
Mots-clés : stable set polytope, Chvátal-rank 
@article{RO_2007__41_3_289_0,
     author = {P\^echer, Arnaud and Wagler, Annegret K.},
     title = {A note on the {Chv\'atal-rank} of clique family inequalities},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {289--294},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {3},
     year = {2007},
     doi = {10.1051/ro:2007022},
     mrnumber = {2348003},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ro:2007022/}
}
TY  - JOUR
AU  - Pêcher, Arnaud
AU  - Wagler, Annegret K.
TI  - A note on the Chvátal-rank of clique family inequalities
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2007
SP  - 289
EP  - 294
VL  - 41
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ro:2007022/
DO  - 10.1051/ro:2007022
LA  - en
ID  - RO_2007__41_3_289_0
ER  - 
%0 Journal Article
%A Pêcher, Arnaud
%A Wagler, Annegret K.
%T A note on the Chvátal-rank of clique family inequalities
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2007
%P 289-294
%V 41
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ro:2007022/
%R 10.1051/ro:2007022
%G en
%F RO_2007__41_3_289_0
Pêcher, Arnaud; Wagler, Annegret K. A note on the Chvátal-rank of clique family inequalities. RAIRO - Operations Research - Recherche Opérationnelle, Tome 41 (2007) no. 3, pp. 289-294. doi : 10.1051/ro:2007022. http://archive.numdam.org/articles/10.1051/ro:2007022/

[1] A. Ben Rebea, Étude des stables dans les graphes quasi-adjoints. Ph.D. thesis, Univ. Grenoble (1981).

[2] W. Cook, R. Kannan and A. Schrijver, Chvátal closures for mixed integer programming problems. Math. Program. 47 (1990) 155-174. | Zbl

[3] M. Chudnovsky and P. Seymour, Claw-free graphs VI. The structure of quasi-line graphs. manuscript (2004).

[4] V. Chvátal, Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Math. 4 (1973) 305-337. | Zbl

[5] V. Chvátal, On certain polytopes associated with graphs 18 (1975) 138-154. | Zbl

[6] V. Chvátal, W. Cook and M. Hartmann, On cutting-plane proofs in combinatorial optimization. Linear Algebra Appl. 114/115 (1989) 455-499. | Zbl

[7] F. Eisenbrand, G. Oriolo, G. Stauffer and P. Ventura, Circular one matrices and the stable set polytope of quasi-line graphs. Lect. Notes Comput. Sci. 3509 (2005) 291-305. | Zbl

[8] 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. | Zbl

[9] R.E. Gomory, Outline of an algorithm for integer solutions to linear programs. Bull. Amer. Math. Soc. 64 (1958) 27-278. | Zbl

[10] T.M. Liebling, G. Oriolo, B. Spille, and G. Stauffer, On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs. Math. Methods Oper. Res. 59 (2004) 25-35 | Zbl

[11] G. Oriolo, On the Stable Set Polytope for Quasi-Line Graphs, Special issue on stability problems. Discrete Appl. Math. 132 (2003) 185-201. | Zbl

[12] A. Pêcher and A. Wagler, Almost all webs are not rank-perfect 105 (2006) 311-328. | Zbl

[13] G. Stauffer, On the Stable Set Polytope of Claw-free Graphs. Ph.D. thesis, EPF Lausanne (2005).

Cité par Sources :