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

Abstract

top

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.


How to cite

top

Pêcher, 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
  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.  Zbl0711.90057
  3. M. Chudnovsky and P. Seymour, Claw-free graphs VI. The structure of quasi-line graphs. manuscript (2004).  Zbl1258.05055
  4. V. Chvátal, Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Math.4 (1973) 305–337.  Zbl0253.05131
  5. V. Chvátal, On certain polytopes associated with graphs. J. Comb. Theory (B)18 (1975) 138–154.  Zbl0277.05139
  6. V. Chvátal, W. Cook and M. Hartmann, On cutting-plane proofs in combinatorial optimization. Linear AlgebraAppl.114/115 (1989) 455–499.  Zbl0676.90059
  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.  Zbl1119.90066
  8. R. Giles and L.E. Trotter Jr., On stable set polyhedra for K1,3-free graphs. J. Comb. Theory B 31 (1981) 313–326.  Zbl0494.05032
  9. R.E. Gomory, Outline of an algorithm for integer solutions to linear programs. Bull. Amer. Math. Soc.64 (1958) 27–278.  Zbl0085.35807
  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 Zbl1060.90079
  11. G. Oriolo, On the Stable Set Polytope for Quasi-Line Graphs, Special issue on stability problems. Discrete Appl. Math.132 (2003) 185–201.  Zbl1052.90108
  12. A. Pêcher and A. Wagler, Almost all webs are not rank-perfect. Math. Program.B105 (2006) 311–328.  Zbl1083.05032
  13. G. Stauffer, On the Stable Set Polytope of Claw-free Graphs. Ph.D. thesis, EPF Lausanne (2005).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.