A note on the Chvátal-rank of clique family inequalities
Arnaud Pêcher; Annegret K. Wagler
RAIRO - Operations Research (2007)
- Volume: 41, Issue: 3, page 289-294
- ISSN: 0399-0559
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topPêcher, Arnaud, and Wagler, Annegret K.. "A note on the Chvátal-rank of clique family inequalities." RAIRO - Operations Research 41.3 (2007): 289-294. <http://eudml.org/doc/250135>.
@article{Pêcher2007,
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.
},
author = {Pêcher, Arnaud, Wagler, Annegret K.},
journal = {RAIRO - Operations Research},
keywords = {Stable set polytope; Chvátal-rank; stable set polytope; Chvàtal-Rank},
language = {eng},
month = {8},
number = {3},
pages = {289-294},
publisher = {EDP Sciences},
title = {A note on the Chvátal-rank of clique family inequalities},
url = {http://eudml.org/doc/250135},
volume = {41},
year = {2007},
}
TY - JOUR
AU - Pêcher, Arnaud
AU - Wagler, Annegret K.
TI - A note on the Chvátal-rank of clique family inequalities
JO - RAIRO - Operations Research
DA - 2007/8//
PB - EDP Sciences
VL - 41
IS - 3
SP - 289
EP - 294
AB -
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.
LA - eng
KW - Stable set polytope; Chvátal-rank; stable set polytope; Chvàtal-Rank
UR - http://eudml.org/doc/250135
ER -
References
top- A. Ben Rebea, Étude des stables dans les graphes quasi-adjoints. Ph.D. thesis, Univ. Grenoble (1981).
- W. Cook, R. Kannan and A. Schrijver, Chvátal closures for mixed integer programming problems. Math. Program.47 (1990) 155–174.
- M. Chudnovsky and P. Seymour, Claw-free graphs VI. The structure of quasi-line graphs. manuscript (2004).
- V. Chvátal, Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Math.4 (1973) 305–337.
- 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 AlgebraAppl.114/115 (1989) 455–499.
- 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.
- R. Giles and L.E. Trotter Jr., On stable set polyhedra for K1,3-free graphs. J. Comb. Theory B 31 (1981) 313–326.
- R.E. Gomory, Outline of an algorithm for integer solutions to linear programs. Bull. Amer. Math. Soc.64 (1958) 27–278.
- 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
- G. Oriolo, On the Stable Set Polytope for Quasi-Line Graphs, Special issue on stability problems. Discrete Appl. Math.132 (2003) 185–201.
- A. Pêcher and A. Wagler, Almost all webs are not rank-perfect. Math. Program.B105 (2006) 311–328.
- G. Stauffer, On the Stable Set Polytope of Claw-free Graphs. Ph.D. thesis, EPF Lausanne (2005).
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