On the Chv´ atal-rank of Antiwebs
Eugenia Holm, Luis M. Torres
1
and Annegret K. Wagler
2
Institute for Mathematical Optimization
Otto-von-Guericke Universit¨ at
Universit¨ atsplatz 2, D-39106, Magdeburg, Germany
Departamento de Matem´atica
Escuela Polit´ ecnica Nacional
Ladr´on de Guevara E11-253, Quito, Ecuador
Abstract
We present an algorithm for computing both upper and lower bounds on the
Chv´ atal-rank of antiwebs, starting from the edge constraint stable set polytope.
With the help of this algorithm we have been able to compute the exact values of
the Chv´atal-rank for all antiwebs containing up to 5,000 nodes. Moreover, the algo-
rithm can be easily adapted to start from the clique constraint stable set polytope.
Keywords: polyhedral combinatorics, stable sets, Chv´ atal-rank
For any polyhedron P ⊂ R
n
, let P
I
denote the convex hull of all integer
points in P . Given a ∈ Z
n
and b ∈ Z, if the inequality a
T
x ≤ b is valid for
P and tight for some x
∗
∈ P , then the inequality a
T
x ≤⌊b⌋ is valid for P
I
,
but violated by x
∗
. Such an inequality is called a Chv´atal-Gomorycut for P .
Define P
′
to be the set of points of P satisfying all Chv´ atal-Gomory cuts for
P , and let P
0
= P and P
t+1
=(P
t
)
′
for non-negative integers t. Obviously
P
I
⊆ P
t
⊆ P holds for every t. Chv´atal[1] showed that for each polyhedron P
1
Email: ltorres@math.epn.edu.ec
2
Email: wagler@mail.math.uni-magdeburg.de
Electronic Notes in Discrete Mathematics 36 (2010) 183–190
1571-0653/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
www.elsevier.com/locate/endm
doi:10.1016/j.endm.2010.05.024