Directory UMM :Data Elmu:jurnal:O:Operations Research Letters:Vol27.Issue4.2000:
Operations Research Letters 27 (2000) 153–161
www.elsevier.com/locate/dsw
New facets for the set packing polytope
Lazaro Canovasa , Mercedes Landeteb , Alfredo Marna; ∗
a Departamento
de Estadstica e Investigacion Operativa, Facultad de Matematicas, Universidad de Murcia, Campus de Espinardo,
30071 Murcia, Spain
b Centro de Investigaci
on Operativa, Universidad Miguel Hernandez, Spain
Received 15 October 1999; received in revised form 8 June 2000; accepted 2 August 2000
Abstract
We introduce a family of graphs, named grilles, and a facet of the set packing polytope associated with a grille. The proof
is based on a new facet generating procedure which is valid in a wider context. We also obtain new facets for the simple
c 2000 Elsevier Science B.V. All rights reserved.
plant location polytope.
Keywords: Polyhedral combinatoric; Facets; Packing; Discrete location
1. Introduction
Throughout this paper it is assumed that the graphs
are nite, without loops, without multiple edges,
undirected and connected. Let G = (V; E) be a graph
with node set V and edge set E. G is said to be odd
(resp. even) if |V | is odd (resp. even). G denotes
the edge-complement of G. The incidence vector
of a subset B of V is a binary vector (x1 ; : : : ; x|V | )
where xj = 1 if and only if the jth node of V belongs to B, j = 1; : : : ; |V |. A nonempty subset of
V of mutually nonadjacent nodes in G is called a
packing (anti-clique, stable set, independent set).
A maximal packing is a packing which is not a
proper subset of another packing. A complete graph
is that in which all the nodes are pairwise adjacent.
A clique in G is a maximal complete subgraph. A
∗ Corresponding author. Fax: +34-68-364-182.
E-mail address: [email protected] (A. Marn).
path (v1 ; e1 ; v2 ; e2 ; : : : ; v‘−1 ; e‘−1 ; v‘ ) is a graph with
distinct nodes {v1 ; : : : ; v‘ } and edges {e1 ; : : : ; e‘−1 }
and such that ei = (vi ; vi+1 ), i = 1; : : : ; ‘ − 1. A cycle (v1 ; e1 ; v2 ; e2 ; : : : ; v‘ ; e‘ ) is a graph with distinct
nodes {v1 ; : : : ; v‘ } and edges {e1 ; : : : ; e‘ } and such
that ei = (vi ; vi+1 ), i = 1; : : : ; ‘ − 1 and e‘ = (v‘ ; v1 ).
A chord for a cycle (v1 ; e1 ; : : : ; v‘ ; e‘ ) is an edge
e 6∈ {e1 ; : : : ; e‘ } linking two nodes in the cycle. A hole
is a chordless cycle with more than three nodes. An
anti-hole is the edge-complement of a hole. A web
W (n; k),S
16k6n=2, is a graph (V; E) such that |V |=n
n
and E = i=1 {(i; i + k); (i; i + k + 1); : : : ; (i; i + k + n)}.
Here n + 1 is identi ed with 1 and so on. An anti-web
is the edge-complement of a web. The neighborhood
N (v) of a node v is the set of nodes that are adjacent
to v. The incidence degree (v) of a node v is the
cardinality of its neighborhood. A star is a connected
graph all whose nodes, except one, have incidence
degree 1. PI (G) is the set of incidence vectors of
all packings of G, and the polytope (polyhedron)
c 2000 Elsevier Science B.V. All rights reserved.
0167-6377/00/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 0 0 ) 0 0 0 5 6 - 0
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L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
associated with G, P(G), is the convex hull of PI (G).
(It holds that P(G) is a full-dimensional polytope, and
a vector x is a vertex of P(G) if and only if x ∈ PI (G)).
A set packing problem is a binary optimization
problem
SPP : Opt{cx: Ax61m ; x ∈ {0; 1}n };
where c ∈ Rn , A ∈ {0; 1}m×n and 1m is an m-vector of
ones. The graph associated with (intersection graph
of) SPP is G = (V; E) with |V | = n and (vi ; vj ) ∈ E
if and only if the ith and jth columns of A are not
orthogonal. Then, if G is the graph associated with
SPP, the feasible set of SPP is PI (G) and the optimal
solutions of SPP can be obtained by solving the linear
optimization problem
Opt{cx: x ∈ P(G)}:
A linear inequality x60 is said to be valid for P(G)
if it holds for all x ∈ P(G) (if and only if it holds for
all x ∈ PI (G)). A valid inequality for P(G) is a facet
of P(G) if and only if it is satis ed as an equality (exactly) by |V | anely independent vertices of P(G).
Apart from the trivial facets −xj 60, all the facets of
P(G) verify j ¿0 ∀j and 0 ¿ 0 (nontrivial facets).
Up to multiplication by a positive constant, there is
a unique set of facets i x60i ; i = 1; : : : ; ‘, such that
P(G) = {x: i x60i ; i = 1; : : : ; ‘}. A set of linear inequalities satisfying the last condition is called a de ning linear system of P(G). Since set packing problems have a large variety of practical applications, and
linear optimization problems are able to be solved by
means of several procedures, it is a matter of interest to contribute to the characterization of the de ning
linear system of P(G), i.e., to obtain facets of P(G).
Sets of nodes will be usually denoted by V or Vi ,
and the same node will be denoted indi erently by vj
and j. In particular, j will be used in the gures, summations and subindices and vj in the text. Frequently,
the expression facet of the graph will be used instead
of facet of the polyhedron associated with the graph,
for brevity.
In order to nd facets of set packing polytopes, it
is useful to identify families of graphs with associated known facets, and to devise methods for transforming a graph and an associated facet into other
pair graph-facet. The seminal papers Balas and Zemel
[1], Chvatal [8], Nemhauser and Trotter [10], Padberg
[11–13] and Trotter [14] gave the bases of the facet
obtaining for set packing problems. One can nd there
the rst families of facet de ning graphs: Cliques, odd
holes, webs, odd anti-holes and anti-webs.
Proposition 1 (Nemhauser and Trotter Jr. [10] and
Padberg [11,13]). Let G = (V; E) beP
a graph and let
B be a subset of V. The inequality j∈B xj 61 is a
facet of P(G) if and only if the subgraph induced by
B is a clique in G.
Consequently, a facet with right-hand side 1 and
binary coecients is called clique facet.
Proposition 2 (Nemhauser and Trotter Jr. [10] and
Padberg [11]). Let
P G = (V; E) be an odd hole. Then;
the inequality
j∈V xj 6(|V | − 1)=2 is a facet of
P(G).
Proposition 3 (Nemhauser and Trotter Jr. [10] and
Padberg [11]). Let GP= (V; E) be an odd anti-hole.
Then; the inequality j∈V xj 62 is a facet of P(G).
Proposition 4 ( Padberg [13] and Trotter [14]). Let
G = W (n; k) be a web. Then;
Pn
1. The inequality j=1 xj 6k is a facet of P(G) if
and only if g:c:d:(n; k) = 1.
2. P
If g:c:d:(n; k) = 1 and k ¿ 1; the inequality
n
j=1 xj 6⌊n=k⌋ is a facet of P(G).
A method for obtaining facets of P(G) from facets
of the polytopes associated with its subgraphs was
simultaneously obtained by several authors ([11] for
odd holes, [10,12,13] for the general case). We will
refer to it as usual lifting procedure.
Proposition 5 (Nemhauser and Trotter [10] and
Padberg [11–13]). Let G P
= (V; E) be a graph with
n−1
|V | = n. If the inequality j=1 j xj 60 i s a facet
of the subgraph of G induced by V − {vn }; the
Pn−1
inequality j=1 j xj + n x n 60 ; where
n = 0
−max
n−1
X
j=1
j xj : x ∈ PI (G); x‘ = 0 ∀‘ ∈ N (vn )
is a facet of P(G).
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L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
More transformations of pairs graph-facet can be
consulted in [13], Cho et al. [7] and Wolsey [15]. Additional families of graphs and associated facets have
been studied in the literature, see [2,5] for wheels,
see [2,3] for series–parallel and other special graphs.
In Canovas et al. [4], new results concerning the facial structure of set packing polyhedra are presented;
in particular, new facet generating methods and facet
de ning graphs are given.
In this paper, a new family of graphs is introduced
and associated facets are obtained. The main result is
presented in Section 3, where the details on the graphs,
named grilles, are given. The main proof is based on
a facet generating method which is shown in Section
2. Finally, in Section 4 we show the usefulness of the
results by obtaining some new facets for the simple
plant location polytope.
2. The facet generating procedure
Proposition 5 is the best-known and most used
method in order to obtain facets of P(G) from
facets of polytopes associated with its subgraphs.
There are similar methods in the literature which
also obtain facets of P(G) from facets of polytopes
associated with its subgraphs but simultaneously lifting several coecients. A di erent approach is to
consider facets associated with graphs of lower dimension, which are not subgraphs of G but can be
obtained from it by means of a given transformation,
and change them into facets of P(G). Wolsey [15]
(Proposition 3) gave one of these methods in which
the “small” graph is obtained by replacing a path
with three nodes {va ; (va ; vb ); vb ; (vb ; vc ); vc } where
N (va ) ∩ N (vc ) = {vb } by a single node linked to all
the nodes in (N (va ) ∪ N (vc )) − {vb }. As shown in
[4], this method can be extended in several ways:
here we give a di erent generalization. It is important
to notice that we start with the “small” graph G and
with a facet of P(G) and give the method to obtain
the “large” graph, which we call G g ; and a facet
of P(G g ):
The following construction is sketched in Fig. 1.
Construction 1. Given a graph G = (V; E); |V | =
n, and a selected node vn ∈ E, a new graph G g is
Fig. 1. Illustration of Construction 1.
obtained by
1. Covering the neighborhood of vn , N (vn ), by means
of m sets S
of nodes Vi ⊂ V , Vi 6= ∅; i = 1; : : : ; m (i.e.
m
N (vn ) = i=1 Vi ).
2. Introducing m new nodes vn+i and linking each
vertex in Vi to vn+i .
3. Linking vn to vn+i , i = 1; : : : ; m, only.
Note that the subsets Vi are not necessarily pairwise
disjoint. Wolsey’s transformation can be obtained as
a particular case when m = 2 and V1 ∩ V2 = ∅, and then
Proposition 3 in [15] is obtained from the following
result.
Pn
Theorem 1. Let j=1 j xj 60 be a nontrivial facet
of P(G) and de ne
n
X
j xj : x ∈ PI (G); x‘ = 0
(Q) := max
j=1
∀v‘ ∈
[
t∈Q
Vt ∪ {vn }
for Q ⊂{1; : : : ; m}. If (Q) = 0 for all Q such that
|Q| = m − 1; then the inequality
n−1
X
j xj + (m − 1)n x n +
j=1
60 + (m − 1)n
is a facet of P(G g ).
m
X
n x n+i
i=1
(1)
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L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
Proof. First, it will be shown that (1) is a valid inequality for P(G g ) and then that n+m anely independent points in PI (G g ) exist satisfying (1) exactly. Note
that, from the assumptions of the theorem, (Q) = 0
holds for all Q ⊂{1; : : : ; m} such that |Q|6m − 1.
Let x1 be a vertex of P(G g ). If x1n = 1, then x1n+i = 0,
i = 1; : : : ; m, and the point x2 given by xj2 = xj1 for
j = 1; : : : ; n − 1, xn2 = 0, is a vertex of P(G), thus
n−1
X
j xj1 + (m − 1)n x1n
j=1
=
n
X
j xj2 + (m − 1)n 60 + (m − 1)n :
j=1
If x1n = 0 and x1n+i = 1, i = 1; : : : ; m, the point x3 given
by xj3 = xj1 for j = 1; : : : ; n − 1, x3n = 1, is a vertex of
P(G) and
n−1
X
j xj1 +
j=1
m
X
Pn
and j=1 j xjn+i = 0 . It can be rightly checked that
the n + m points given by
k
s
{Y k }n+m
k=1 := {(X ; 1; 0; 0; : : : ; 0)}k=1 ;
{(X k ; 0; 1; 1; : : : ; 1)}nk=s+1 ; {(X n+i ; 0; H i )}m
i=1 ;
where H i = (1; 1; : : : ; 1; 0; 1; : : : ; 1; 1) is an m-vector of
ones with a zero in the ith position, are vertices of
P(G g ) and satisfy (1) exactly. Moreover, the n+m−1
rst points are clearly a set of anely independent
points of Rn+m , and assuming
Y n+m =
k Y k +
k=1
m−1
X
k Y n+k
k=1
Ps
Pn
it follows k=1 k = 0, k=s+1 k = m − 1, k = −1
Pn
Pm−1
for k =1; : : : ; m−1, X n+m = k=1 k X k − k=1 X n+k
and
0 = ({1; : : : ; m − 1}) =
n x1n+i
n
X
j xj3 − n + mn 60 + (m − 1)n :
=
If x1n = 0; x1n+i = 1 for, say, i = 1; : : : ; q; 16q6m − 1,
and x1n+i = 0, for i = q + 1; : : : ; m, using the above
de ned point x2 it holds
=
n
X
j=1
j xj2 =
n−1
X
j xj1 ;
j xjn+m
k
s
X
k xjk
−
k=1
n−1
X
j xjk −
j=1
m−1
X
xjn+k
k=1
n−1
m−1
XX
!
j xjn+k
k=1 j=1
k (0 − n ) +
n
X
k=s+1
k 0 −
m−1
X
0 = 0;
k=1
a contradiction.
m
X
n x1n+i
i=1
n−1
X
n
X
n
X
k=1
j=1
j xj1 + (m − 1)n x1n +
=
j
k=1
=
implying
j=1
n−1
X
j=1
j=1
0 ¿
n−1
X
j=1
i=1
=
n−1
X
n
X
j xj1 + qn 60 + qn 60 + (m − 1)n :
j=1
Example 1. Let G be the left-hand graph of Fig. 2
vn = v15 . Set m = 4, V1 = {1; 2}, V2 = {2; 3}, V3 = {4}
and V4 = {5}. Then, G g is the right-hand graph of
Fig. 2. Consider the facet of P(G)
x =
Now, let {(X k ; 0)}sk=1 , {(X k ; 1)}nk=s+1
P,nbe n anely
independent points of PI (G) satisfying j=1 j xj =0 ,
and let (X n+i ; 0), i = 1; : : : ; m, be m vertices of P(G)
satisfying
[
Vt ∪ {vn }
x‘n+i = 0 ∀v‘ ∈
t∈{1;:::; m}−{i}
8
X
xj + 3x9 + x10 + x11 + 2x12 + x13 + x14
j=1
+ x15 67:
(2)
In order to apply Theorem 1, it should be checked
that, after deleting v15 and any union of three sets
Vi from G, a packing which satis es (2) exactly
remains. The packings {v5 ; v6 ; v8 ; v10 ; v11 ; v13 ; v14 },
{v4 ; v7 ; v9 ; v12 }, {v3 ; v7 ; v9 ; v12 } and {v1 ; v7 ; v9 ; v12 }
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L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
Fig. 2. Illustration of Example 1.
satisfy (2) exactly and do not contain any node
in V1 ∪ V2 ∪ V3 ∪ {v15 }, V1 ∪ V2 ∪ V4 ∪ {v15 },
V1 ∪ V3 ∪ V4 ∪ {v15 } and V2 ∪ V3 ∪ V4 ∪ {v15 }, respectively. Therefore, the following facet of P(G g ) is
obtained:
8
X
xj + 3x9 + x10 + x11 + 2x12 + x13 + x14 + 3x15
j=1
+ x16 + x17 + x18 + x19 610:
3. Grilles
This section is devoted to the construction of a new
family of facet de ning graphs which we call grilles
and its associated facet.
Construction 2. A grille is obtained by
1. Considering p¿3 stars with at least three nodes
each; the nodes with degree greater than one will be
called interior nodes and numbered {v1 ; : : : ; vp };
the nodes with degree one will be called exterior
nodes associated with an interior node (the neighbor), and numbered {vp+1 ; : : : ; vn }.
2. Linking the exterior nodes to one another so that
(a) two exterior nodes are not linked if they have
the same associated interior node,
(b) all the nodes have incidence degree at least
two,
(c) given two interior nodes, there exists a
three-edges path between them,
(d) given an exterior node ve associated with the
interior node vi , another interior node vj , j ∈
{1; : : : ; p}−{i} exists so that N (ve )∩N (vj ) 6=
∅ and N (v‘ ) ∩ N (vj ) = ∅ ∀v‘ ∈ N (vi ) − {ve }.
Note that the last condition limits the number of
nodes of the stars.
Example 2. Fig. 3(a) shows the minimum grille,
which is also an odd hole. Graphs 3(b) and 3(c) are
both grilles based on the same stars; 3(b) has the
minimum set of edges which satis es condition 2(c),
whereas 3(c) has a maximal set of edges which satis es condition 2(d). Graph 3(d) is based on ve stars,
two of them with four nodes; the number on an exterior node indicates a possible value for j in condition
2(d) of the Construction 2.
Theorem 2. The inequality
p
X
j=1
((vj ) − 1) xj +
n
X
j=p+1
xj 6
p
X
((vj ) − 1) + 1
j=1
(3)
de nes a facet of the set packing polytope associated
with the grille with interior nodes {v1 ; : : : ; vp } and
exterior nodes {vp+1 ; : : : ; vn }.
Proof. Let G = (VG ; EG ) be a grille with interior
nodes {v1 ; : : : ; vp } and exterior nodes {vp+1 ; : : : ; vn }.
We shall show by induction that, applying Theorem 1
in a certain way to the complete graph Q with nodes
: ; vp }, p times, (i) Q becomes G and (ii) the
{v1 ; : :P
p
facet j=1 xj 61 of P(Q) results in the facet (3) of
P(G).
Let Q be the 0th intermediate graph. Assume that
the (k − 1)th intermediate graph, k = 1; : : : ; p, consists
of the rst k − 1 stars of the grille (with interior nodes
v1 ; : : : ; vk−1 and a set of exterior nodes Wk−1 ), a clique
with p−k +1 nodes (vk ; : : : ; vp ), a set of edges linking
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L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
Fig. 3. Illustration of Example 2. Black- lled nodes are interior nodes, thick edges correspond with stars.
the exterior nodes of the stars like in the grille, and
more edges linking the exterior nodes of the stars to
the nodes in the clique, according with the following
rule: An exterior node v is linked to a node vh in
the clique if and only if there exists an edge in G
linking the exterior node v to any exterior node of the
star associated with the interior node vh . Moreover,
assume that
k−1
X
((vj ) − 1)xj +
j∈Wk−1
j=1
6
X
k−1
X
((vj ) − 1) + 1
xj +
p
X
xj
j=k
(4)
j=1
is a facet of this intermediate graph. It can be easily
checked that Q satis es all these assumptions.
Then, the kth intermediate graph (k = 1; : : : ; p) is
obtained by (i) taking the (k −1)th intermediate graph
as original graph, (4) as original facet and vk as selected node, (ii) setting m equal to the number of exterior nodes of the kth star in the grille, (iii) choosing
Vi , i = 1; : : : ; m, equal to the set of nodes that are either
(a) exterior nodes associated with v1 ; : : : ; vk−1 which
are linked, in G, to the ith exterior node associated
with vk or (b) nodes of the set {vt : t¿k + 1} such that
a link between an exterior node associated with them
and the ith exterior node associated with vk exists in
G, and (iv) applying Theorem 1 if the condition holds.
As a consequence of the construction, the kth intermediate graph consists of the k rst stars of the grille
(with interior nodes v1 ; : : : ; vk and a set of exterior
nodes Wk ), a clique with p − k nodes (vk+1 ; : : : ; vp ),
a set of edges linking the exterior nodes of the stars
like in the grille, and more edges linking the exterior
nodes of the stars to the nodes in the clique as above.
Moreover, the facet which results from (4) is
k
X
((vj ) − 1)xj +
xj +
j∈Wk
j=1
6
X
k
X
j=1
((vj ) − 1) + 1:
p
X
j=k+1
xj
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L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
Then, after p steps, inequality (3) should be reached
and the proof should be complete.
To end the proof, it is necessary to check that the
condition of Theorem 1 is satis ed in the kth step,
k = 1; : : : ; p. But note that Condition 2(d) of the Construction 2 guarantees that the ith exterior node associated with vk is either the only one linked to a star
or the only one linked to a node in the clique. If it is
a star, say the rst one, its exterior nodes along with
nodes v2 ; : : : ; vk−1 constitute the packing which is necessary to satisfy the condition. If it is a node, say vp ,
the packing includes vp plus the nodes v1 ; : : : ; vk−1 . In
any case, Theorem 1 can be applied and the proof is
complete.
4. New facets of the simple plant location polytope
Grilles are not an entelechy, but subgraphs of
well-known families of graphs can be identi ed as
grilles. In this section we obtain facets for a polytope
associated with the simple plant location problem,
based on grilles. For details about this problem, see
[6]. The interested reader will nd several families of
facets in [7] and Cornuejols and Thizy [9].
The simple plant location problem with p plants
and d destinations can be formulated as a set packing
problem with constraints
p
X
xij 61 ∀j ∈ {1; : : : ; d};
i=1
xij + yi 61 ∀i ∈ {1; : : : ; p}; ∀j ∈ {1; : : : ; d}:
We call G(p; d) the intersection graph associated
with this set of constraints, and label each node with
the name of its associated variable. Fig. 4 shows
G(4; 6).
Theorem 3. The subgraph of G(p; d) induced by a
set of nodes associated with a set of y-variables Qy
and a set of x-variables Qx such that
(i) yi ∈ Qy ∀i ∈ {1; : : : ; p};
(ii) ∀i1 ; i2 ∈ {1; : : : ; p} ∃j ∈ {1; : : : ; d} such that
xi1 j ; xi2 j ∈ Qx ;
(iii) ∀xij ∈ Qx ∃x‘j ∈ Qx ; ‘ 6= i; such that (xit ∈ Qx ⇒
x‘t ∈= Qx ∀t 6= j)
is a grille with set of interior nodes Qy and set of
exterior nodes Qx .
Fig. 4. Illustration of Example 3. All the edges of the complete
graph of 4 nodes have been separately considered.
Proof. It is clear that the subsets of nodes {yi }∪N (yi )
induce disjoint stars. Condition 2(a) of Construction
2 holds because two nodes xij1 ; xij2 are never linked,
2(b) follows from (ii) and (iii), 2(c) follows from (ii)
and 2(d) follows from (iii).
Corollary 1. Given G(p; d) and two sets of variables
Qx and Qy satisfying the conditions given in Theorem
3; the inequality
X
X
X
xij +
(
i − 1)yi 6
(
i − 1) + 1;
(5)
Qx
Qy
Qy
where
i := |{xij : xij ∈ Qx }|; is a facet of P(G(p′ ; d′ ))
for any p′ ¿p; d′ ¿d.
Proof. From Theorem 2, (5) is a facet of the subgraph
induced by Qx ∪ Qy . It is sucient to prove that any
other node in G(p′ ; d′ ) can be lifted with coecient
0 by means of Proposition 5.
Consider one of such nodes v, add it to a grille and
apply Proposition 5 (identifying v with vn ). There are
two possibilities:
1. v is not linked, in G(p′ ; d′ ), to any node of Qx ∪Qy ;
then, it is clear that n takes value 0.
2. v is a node xij with yi ∈ Qy ; then, v is linked
to yi and to the (maybe empty) clique of nodes
{xtj : xtj ∈ Qx }. In this case, a packing exists in
the grille that satis es (5) exactly and does not
contain any of the nodes linked to v. It is given by
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L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
Fig. 5. Illustration of Example 4.
{y‘ : ‘ 6= i} ∪ {xit : xit ∈ Qx }. Therefore, n = 0 and
the proof is complete.
Example 3. The conditions of Theorem 3 indicate that, in order to construct a grille facet for
G(p; d), a complete graph of p∗ 6p nodes should be
edge-covered by a set of d∗ 6d complete subgraphs
in such a way that each node in each subgraph is the
end-point of an edge that is not in another subgraph.
In Fig. 4, the complete 4-graph has been decomposed
into single edges (thick lines). Consider the set
R = { (3; 1); (4; 1); (1; 2); (3; 2); (1; 3); (4; 3); (2; 4);
(3; 4); (2; 5); (4; 5); (1; 6); (2; 6) }:
The inequality
4
X
i=1
2yi +
X
xij 69
(6)
(i; j)∈R
is a facet of all the location polytopes with at least
4 plants and 6 destinations. Note that the y-nodes
(plants) have been numbered in the gure and the
x-cliques (destinations) have been sorted from left to
right.
Now, we show some fractional extreme points of
the polytope associated with the linear relaxation of
SPLP with p = 4 and d = 6 that are cut o by facet
(6). The rst point is given by
(
1=2 if (i; j) ∈ R;
2
1
1
1
1
y1 = y2 = y3 = y4 = 1=2; xij =
0 otherwise:
The second point is
y12 = 1;
y22 = y32 = y42 = 12 ;
2
2
2
2
2
2
2
2
2
= x41
= x32
= x43
= x24
= x34
= x25
= x45
= x26
x31
=
1
2
and xij2 = 0 otherwise. By symmetry, three more points
can be obtained from the latter. All of them are extreme points because the nodes associated with variables taking the value 1=2 induce connected subgraphs
containing an odd cycle, see [10]. It was also proved in
the cited paper that all the extreme points of the linear
relaxation of SPLP are cut o by clique and odd hole
inequalities. The following is an extreme point of the
polytope obtained when all the odd holes are added to
the linear relaxation of SPLP with p = 4 and d = 6:
(
1=3 if (i; j) ∈ R;
3
3
3
3
3
y1 = y2 = y3 = y4 = 2=3; xij =
0 otherwise:
The reader should check that this point is cut o by (6).
Of course, these points would be optimal solutions of
the linear relaxation of SPLP if the objective function
be similar enough to the left-hand side of the facet.
For instance, (x1 ; y1 ) would be the optimal solution
(maximum) if the objective function had the form
4
X
i=1
fi yi +
4 X
6
X
i=1 j=1
bij xij
L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
with
fi = 200 ∀i;
bij =
(
100 if (i; j) ∈ R;
30 otherwise:
Example 4. In Fig. 5, a grille in G(7; 6) is shown
(thick lines), with associated facet
3y1 + 2y2 + y3 + 3y4 + 2y5 + 2y6 + y7
X
+
xij 615:
(7)
Qx
The black- lled node and the thin edges do not belong to the grille, but they illustrate the lifting process
of Corollary 1. The numbers out of the nodes are the
coecients of the nodes which constitute the packing
satisfying (7) exactly and do not belong to the neighborhood of the black- lled node.
References
[1] E. Balas, E. Zemel, Critical cutsets of graphs and canonical
facets of set-packing polytopes, Math. Oper. Res. 2 (1977)
15–19.
[2] F. Barahona, A.R. Mahjoub, Compositions of graphs and
polyhedra II: Stable sets, SIAM J. Discrete Math. 7 (1994)
359–371.
161
[3] F. Barahona, A.R. Mahjoub, Compositions of graphs and
polyhedra III: Graphs with no W4 minor, SIAM J. Discrete
Math. 7 (1994) 372–389.
[4] L. Canovas, M. Landete, A. Marn, Facet obtaining procedures
for set packing problems, Working paper 4=99, Departamento
de Estadstica e Investigacion Operativa, Universidad de
Murcia, 1999.
[5] E. Cheng, W.H. Cunningham, Wheel inequalities for stable
set polytopes, Math. Programming 77 (1997) 389–421.
[6] D.C. Cho, E.L. Johnson, M.W. Padberg, M. R. Rao, On the
uncapacitated plant location problem I: valid inequalities and
facets, Math. Oper. Res. 8 (1983) 579–589.
[7] D.C. Cho, M.W. Padberg, M.R. Rao, On the uncapacitated
plant location problem II: Facets and lifting theorems, Math.
Oper. Res. 8 (1983) 590–612.
[8] V. Chvatal, On certain polytopes associated with graphs, J.
Combin. Theory Ser. B 18 (1975) 138–154.
[9] G. Cornuejols, J.-M. Thizy, Some facets of the simple plant
location polytope, Math. Programming 23 (1982) 50–74.
[10] G.L. Nemhauser, L.E. Trotter Jr., Properties of vertex packing
and independence system polyhedra, Math. Programming 6
(1974) 48–61.
[11] M.W. Padberg, On the facial structure of set packing
polyhedra, Math. Programming 5 (1973) 199–215.
[12] M.W. Padberg, A note on zero-one programming, Oper. Res.
23 (1975) 833–837.
[13] M.W. Padberg, On the complexity of set packing polyhedra,
Ann. Discrete Math. 1 (1977) 421–434.
[14] L.E. Trotter, A class of facet producing graphs for vertex
packing polyhedra, Discrete Math. 12 (1975) 373–388.
[15] L.A. Wolsey, Further facet generating procedures for vertex
packing polytopes, Math. Programming 11 (1976) 158–163.
www.elsevier.com/locate/dsw
New facets for the set packing polytope
Lazaro Canovasa , Mercedes Landeteb , Alfredo Marna; ∗
a Departamento
de Estadstica e Investigacion Operativa, Facultad de Matematicas, Universidad de Murcia, Campus de Espinardo,
30071 Murcia, Spain
b Centro de Investigaci
on Operativa, Universidad Miguel Hernandez, Spain
Received 15 October 1999; received in revised form 8 June 2000; accepted 2 August 2000
Abstract
We introduce a family of graphs, named grilles, and a facet of the set packing polytope associated with a grille. The proof
is based on a new facet generating procedure which is valid in a wider context. We also obtain new facets for the simple
c 2000 Elsevier Science B.V. All rights reserved.
plant location polytope.
Keywords: Polyhedral combinatoric; Facets; Packing; Discrete location
1. Introduction
Throughout this paper it is assumed that the graphs
are nite, without loops, without multiple edges,
undirected and connected. Let G = (V; E) be a graph
with node set V and edge set E. G is said to be odd
(resp. even) if |V | is odd (resp. even). G denotes
the edge-complement of G. The incidence vector
of a subset B of V is a binary vector (x1 ; : : : ; x|V | )
where xj = 1 if and only if the jth node of V belongs to B, j = 1; : : : ; |V |. A nonempty subset of
V of mutually nonadjacent nodes in G is called a
packing (anti-clique, stable set, independent set).
A maximal packing is a packing which is not a
proper subset of another packing. A complete graph
is that in which all the nodes are pairwise adjacent.
A clique in G is a maximal complete subgraph. A
∗ Corresponding author. Fax: +34-68-364-182.
E-mail address: [email protected] (A. Marn).
path (v1 ; e1 ; v2 ; e2 ; : : : ; v‘−1 ; e‘−1 ; v‘ ) is a graph with
distinct nodes {v1 ; : : : ; v‘ } and edges {e1 ; : : : ; e‘−1 }
and such that ei = (vi ; vi+1 ), i = 1; : : : ; ‘ − 1. A cycle (v1 ; e1 ; v2 ; e2 ; : : : ; v‘ ; e‘ ) is a graph with distinct
nodes {v1 ; : : : ; v‘ } and edges {e1 ; : : : ; e‘ } and such
that ei = (vi ; vi+1 ), i = 1; : : : ; ‘ − 1 and e‘ = (v‘ ; v1 ).
A chord for a cycle (v1 ; e1 ; : : : ; v‘ ; e‘ ) is an edge
e 6∈ {e1 ; : : : ; e‘ } linking two nodes in the cycle. A hole
is a chordless cycle with more than three nodes. An
anti-hole is the edge-complement of a hole. A web
W (n; k),S
16k6n=2, is a graph (V; E) such that |V |=n
n
and E = i=1 {(i; i + k); (i; i + k + 1); : : : ; (i; i + k + n)}.
Here n + 1 is identi ed with 1 and so on. An anti-web
is the edge-complement of a web. The neighborhood
N (v) of a node v is the set of nodes that are adjacent
to v. The incidence degree (v) of a node v is the
cardinality of its neighborhood. A star is a connected
graph all whose nodes, except one, have incidence
degree 1. PI (G) is the set of incidence vectors of
all packings of G, and the polytope (polyhedron)
c 2000 Elsevier Science B.V. All rights reserved.
0167-6377/00/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 0 0 ) 0 0 0 5 6 - 0
154
L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
associated with G, P(G), is the convex hull of PI (G).
(It holds that P(G) is a full-dimensional polytope, and
a vector x is a vertex of P(G) if and only if x ∈ PI (G)).
A set packing problem is a binary optimization
problem
SPP : Opt{cx: Ax61m ; x ∈ {0; 1}n };
where c ∈ Rn , A ∈ {0; 1}m×n and 1m is an m-vector of
ones. The graph associated with (intersection graph
of) SPP is G = (V; E) with |V | = n and (vi ; vj ) ∈ E
if and only if the ith and jth columns of A are not
orthogonal. Then, if G is the graph associated with
SPP, the feasible set of SPP is PI (G) and the optimal
solutions of SPP can be obtained by solving the linear
optimization problem
Opt{cx: x ∈ P(G)}:
A linear inequality x60 is said to be valid for P(G)
if it holds for all x ∈ P(G) (if and only if it holds for
all x ∈ PI (G)). A valid inequality for P(G) is a facet
of P(G) if and only if it is satis ed as an equality (exactly) by |V | anely independent vertices of P(G).
Apart from the trivial facets −xj 60, all the facets of
P(G) verify j ¿0 ∀j and 0 ¿ 0 (nontrivial facets).
Up to multiplication by a positive constant, there is
a unique set of facets i x60i ; i = 1; : : : ; ‘, such that
P(G) = {x: i x60i ; i = 1; : : : ; ‘}. A set of linear inequalities satisfying the last condition is called a de ning linear system of P(G). Since set packing problems have a large variety of practical applications, and
linear optimization problems are able to be solved by
means of several procedures, it is a matter of interest to contribute to the characterization of the de ning
linear system of P(G), i.e., to obtain facets of P(G).
Sets of nodes will be usually denoted by V or Vi ,
and the same node will be denoted indi erently by vj
and j. In particular, j will be used in the gures, summations and subindices and vj in the text. Frequently,
the expression facet of the graph will be used instead
of facet of the polyhedron associated with the graph,
for brevity.
In order to nd facets of set packing polytopes, it
is useful to identify families of graphs with associated known facets, and to devise methods for transforming a graph and an associated facet into other
pair graph-facet. The seminal papers Balas and Zemel
[1], Chvatal [8], Nemhauser and Trotter [10], Padberg
[11–13] and Trotter [14] gave the bases of the facet
obtaining for set packing problems. One can nd there
the rst families of facet de ning graphs: Cliques, odd
holes, webs, odd anti-holes and anti-webs.
Proposition 1 (Nemhauser and Trotter Jr. [10] and
Padberg [11,13]). Let G = (V; E) beP
a graph and let
B be a subset of V. The inequality j∈B xj 61 is a
facet of P(G) if and only if the subgraph induced by
B is a clique in G.
Consequently, a facet with right-hand side 1 and
binary coecients is called clique facet.
Proposition 2 (Nemhauser and Trotter Jr. [10] and
Padberg [11]). Let
P G = (V; E) be an odd hole. Then;
the inequality
j∈V xj 6(|V | − 1)=2 is a facet of
P(G).
Proposition 3 (Nemhauser and Trotter Jr. [10] and
Padberg [11]). Let GP= (V; E) be an odd anti-hole.
Then; the inequality j∈V xj 62 is a facet of P(G).
Proposition 4 ( Padberg [13] and Trotter [14]). Let
G = W (n; k) be a web. Then;
Pn
1. The inequality j=1 xj 6k is a facet of P(G) if
and only if g:c:d:(n; k) = 1.
2. P
If g:c:d:(n; k) = 1 and k ¿ 1; the inequality
n
j=1 xj 6⌊n=k⌋ is a facet of P(G).
A method for obtaining facets of P(G) from facets
of the polytopes associated with its subgraphs was
simultaneously obtained by several authors ([11] for
odd holes, [10,12,13] for the general case). We will
refer to it as usual lifting procedure.
Proposition 5 (Nemhauser and Trotter [10] and
Padberg [11–13]). Let G P
= (V; E) be a graph with
n−1
|V | = n. If the inequality j=1 j xj 60 i s a facet
of the subgraph of G induced by V − {vn }; the
Pn−1
inequality j=1 j xj + n x n 60 ; where
n = 0
−max
n−1
X
j=1
j xj : x ∈ PI (G); x‘ = 0 ∀‘ ∈ N (vn )
is a facet of P(G).
155
L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
More transformations of pairs graph-facet can be
consulted in [13], Cho et al. [7] and Wolsey [15]. Additional families of graphs and associated facets have
been studied in the literature, see [2,5] for wheels,
see [2,3] for series–parallel and other special graphs.
In Canovas et al. [4], new results concerning the facial structure of set packing polyhedra are presented;
in particular, new facet generating methods and facet
de ning graphs are given.
In this paper, a new family of graphs is introduced
and associated facets are obtained. The main result is
presented in Section 3, where the details on the graphs,
named grilles, are given. The main proof is based on
a facet generating method which is shown in Section
2. Finally, in Section 4 we show the usefulness of the
results by obtaining some new facets for the simple
plant location polytope.
2. The facet generating procedure
Proposition 5 is the best-known and most used
method in order to obtain facets of P(G) from
facets of polytopes associated with its subgraphs.
There are similar methods in the literature which
also obtain facets of P(G) from facets of polytopes
associated with its subgraphs but simultaneously lifting several coecients. A di erent approach is to
consider facets associated with graphs of lower dimension, which are not subgraphs of G but can be
obtained from it by means of a given transformation,
and change them into facets of P(G). Wolsey [15]
(Proposition 3) gave one of these methods in which
the “small” graph is obtained by replacing a path
with three nodes {va ; (va ; vb ); vb ; (vb ; vc ); vc } where
N (va ) ∩ N (vc ) = {vb } by a single node linked to all
the nodes in (N (va ) ∪ N (vc )) − {vb }. As shown in
[4], this method can be extended in several ways:
here we give a di erent generalization. It is important
to notice that we start with the “small” graph G and
with a facet of P(G) and give the method to obtain
the “large” graph, which we call G g ; and a facet
of P(G g ):
The following construction is sketched in Fig. 1.
Construction 1. Given a graph G = (V; E); |V | =
n, and a selected node vn ∈ E, a new graph G g is
Fig. 1. Illustration of Construction 1.
obtained by
1. Covering the neighborhood of vn , N (vn ), by means
of m sets S
of nodes Vi ⊂ V , Vi 6= ∅; i = 1; : : : ; m (i.e.
m
N (vn ) = i=1 Vi ).
2. Introducing m new nodes vn+i and linking each
vertex in Vi to vn+i .
3. Linking vn to vn+i , i = 1; : : : ; m, only.
Note that the subsets Vi are not necessarily pairwise
disjoint. Wolsey’s transformation can be obtained as
a particular case when m = 2 and V1 ∩ V2 = ∅, and then
Proposition 3 in [15] is obtained from the following
result.
Pn
Theorem 1. Let j=1 j xj 60 be a nontrivial facet
of P(G) and de ne
n
X
j xj : x ∈ PI (G); x‘ = 0
(Q) := max
j=1
∀v‘ ∈
[
t∈Q
Vt ∪ {vn }
for Q ⊂{1; : : : ; m}. If (Q) = 0 for all Q such that
|Q| = m − 1; then the inequality
n−1
X
j xj + (m − 1)n x n +
j=1
60 + (m − 1)n
is a facet of P(G g ).
m
X
n x n+i
i=1
(1)
156
L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
Proof. First, it will be shown that (1) is a valid inequality for P(G g ) and then that n+m anely independent points in PI (G g ) exist satisfying (1) exactly. Note
that, from the assumptions of the theorem, (Q) = 0
holds for all Q ⊂{1; : : : ; m} such that |Q|6m − 1.
Let x1 be a vertex of P(G g ). If x1n = 1, then x1n+i = 0,
i = 1; : : : ; m, and the point x2 given by xj2 = xj1 for
j = 1; : : : ; n − 1, xn2 = 0, is a vertex of P(G), thus
n−1
X
j xj1 + (m − 1)n x1n
j=1
=
n
X
j xj2 + (m − 1)n 60 + (m − 1)n :
j=1
If x1n = 0 and x1n+i = 1, i = 1; : : : ; m, the point x3 given
by xj3 = xj1 for j = 1; : : : ; n − 1, x3n = 1, is a vertex of
P(G) and
n−1
X
j xj1 +
j=1
m
X
Pn
and j=1 j xjn+i = 0 . It can be rightly checked that
the n + m points given by
k
s
{Y k }n+m
k=1 := {(X ; 1; 0; 0; : : : ; 0)}k=1 ;
{(X k ; 0; 1; 1; : : : ; 1)}nk=s+1 ; {(X n+i ; 0; H i )}m
i=1 ;
where H i = (1; 1; : : : ; 1; 0; 1; : : : ; 1; 1) is an m-vector of
ones with a zero in the ith position, are vertices of
P(G g ) and satisfy (1) exactly. Moreover, the n+m−1
rst points are clearly a set of anely independent
points of Rn+m , and assuming
Y n+m =
k Y k +
k=1
m−1
X
k Y n+k
k=1
Ps
Pn
it follows k=1 k = 0, k=s+1 k = m − 1, k = −1
Pn
Pm−1
for k =1; : : : ; m−1, X n+m = k=1 k X k − k=1 X n+k
and
0 = ({1; : : : ; m − 1}) =
n x1n+i
n
X
j xj3 − n + mn 60 + (m − 1)n :
=
If x1n = 0; x1n+i = 1 for, say, i = 1; : : : ; q; 16q6m − 1,
and x1n+i = 0, for i = q + 1; : : : ; m, using the above
de ned point x2 it holds
=
n
X
j=1
j xj2 =
n−1
X
j xj1 ;
j xjn+m
k
s
X
k xjk
−
k=1
n−1
X
j xjk −
j=1
m−1
X
xjn+k
k=1
n−1
m−1
XX
!
j xjn+k
k=1 j=1
k (0 − n ) +
n
X
k=s+1
k 0 −
m−1
X
0 = 0;
k=1
a contradiction.
m
X
n x1n+i
i=1
n−1
X
n
X
n
X
k=1
j=1
j xj1 + (m − 1)n x1n +
=
j
k=1
=
implying
j=1
n−1
X
j=1
j=1
0 ¿
n−1
X
j=1
i=1
=
n−1
X
n
X
j xj1 + qn 60 + qn 60 + (m − 1)n :
j=1
Example 1. Let G be the left-hand graph of Fig. 2
vn = v15 . Set m = 4, V1 = {1; 2}, V2 = {2; 3}, V3 = {4}
and V4 = {5}. Then, G g is the right-hand graph of
Fig. 2. Consider the facet of P(G)
x =
Now, let {(X k ; 0)}sk=1 , {(X k ; 1)}nk=s+1
P,nbe n anely
independent points of PI (G) satisfying j=1 j xj =0 ,
and let (X n+i ; 0), i = 1; : : : ; m, be m vertices of P(G)
satisfying
[
Vt ∪ {vn }
x‘n+i = 0 ∀v‘ ∈
t∈{1;:::; m}−{i}
8
X
xj + 3x9 + x10 + x11 + 2x12 + x13 + x14
j=1
+ x15 67:
(2)
In order to apply Theorem 1, it should be checked
that, after deleting v15 and any union of three sets
Vi from G, a packing which satis es (2) exactly
remains. The packings {v5 ; v6 ; v8 ; v10 ; v11 ; v13 ; v14 },
{v4 ; v7 ; v9 ; v12 }, {v3 ; v7 ; v9 ; v12 } and {v1 ; v7 ; v9 ; v12 }
157
L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
Fig. 2. Illustration of Example 1.
satisfy (2) exactly and do not contain any node
in V1 ∪ V2 ∪ V3 ∪ {v15 }, V1 ∪ V2 ∪ V4 ∪ {v15 },
V1 ∪ V3 ∪ V4 ∪ {v15 } and V2 ∪ V3 ∪ V4 ∪ {v15 }, respectively. Therefore, the following facet of P(G g ) is
obtained:
8
X
xj + 3x9 + x10 + x11 + 2x12 + x13 + x14 + 3x15
j=1
+ x16 + x17 + x18 + x19 610:
3. Grilles
This section is devoted to the construction of a new
family of facet de ning graphs which we call grilles
and its associated facet.
Construction 2. A grille is obtained by
1. Considering p¿3 stars with at least three nodes
each; the nodes with degree greater than one will be
called interior nodes and numbered {v1 ; : : : ; vp };
the nodes with degree one will be called exterior
nodes associated with an interior node (the neighbor), and numbered {vp+1 ; : : : ; vn }.
2. Linking the exterior nodes to one another so that
(a) two exterior nodes are not linked if they have
the same associated interior node,
(b) all the nodes have incidence degree at least
two,
(c) given two interior nodes, there exists a
three-edges path between them,
(d) given an exterior node ve associated with the
interior node vi , another interior node vj , j ∈
{1; : : : ; p}−{i} exists so that N (ve )∩N (vj ) 6=
∅ and N (v‘ ) ∩ N (vj ) = ∅ ∀v‘ ∈ N (vi ) − {ve }.
Note that the last condition limits the number of
nodes of the stars.
Example 2. Fig. 3(a) shows the minimum grille,
which is also an odd hole. Graphs 3(b) and 3(c) are
both grilles based on the same stars; 3(b) has the
minimum set of edges which satis es condition 2(c),
whereas 3(c) has a maximal set of edges which satis es condition 2(d). Graph 3(d) is based on ve stars,
two of them with four nodes; the number on an exterior node indicates a possible value for j in condition
2(d) of the Construction 2.
Theorem 2. The inequality
p
X
j=1
((vj ) − 1) xj +
n
X
j=p+1
xj 6
p
X
((vj ) − 1) + 1
j=1
(3)
de nes a facet of the set packing polytope associated
with the grille with interior nodes {v1 ; : : : ; vp } and
exterior nodes {vp+1 ; : : : ; vn }.
Proof. Let G = (VG ; EG ) be a grille with interior
nodes {v1 ; : : : ; vp } and exterior nodes {vp+1 ; : : : ; vn }.
We shall show by induction that, applying Theorem 1
in a certain way to the complete graph Q with nodes
: ; vp }, p times, (i) Q becomes G and (ii) the
{v1 ; : :P
p
facet j=1 xj 61 of P(Q) results in the facet (3) of
P(G).
Let Q be the 0th intermediate graph. Assume that
the (k − 1)th intermediate graph, k = 1; : : : ; p, consists
of the rst k − 1 stars of the grille (with interior nodes
v1 ; : : : ; vk−1 and a set of exterior nodes Wk−1 ), a clique
with p−k +1 nodes (vk ; : : : ; vp ), a set of edges linking
158
L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
Fig. 3. Illustration of Example 2. Black- lled nodes are interior nodes, thick edges correspond with stars.
the exterior nodes of the stars like in the grille, and
more edges linking the exterior nodes of the stars to
the nodes in the clique, according with the following
rule: An exterior node v is linked to a node vh in
the clique if and only if there exists an edge in G
linking the exterior node v to any exterior node of the
star associated with the interior node vh . Moreover,
assume that
k−1
X
((vj ) − 1)xj +
j∈Wk−1
j=1
6
X
k−1
X
((vj ) − 1) + 1
xj +
p
X
xj
j=k
(4)
j=1
is a facet of this intermediate graph. It can be easily
checked that Q satis es all these assumptions.
Then, the kth intermediate graph (k = 1; : : : ; p) is
obtained by (i) taking the (k −1)th intermediate graph
as original graph, (4) as original facet and vk as selected node, (ii) setting m equal to the number of exterior nodes of the kth star in the grille, (iii) choosing
Vi , i = 1; : : : ; m, equal to the set of nodes that are either
(a) exterior nodes associated with v1 ; : : : ; vk−1 which
are linked, in G, to the ith exterior node associated
with vk or (b) nodes of the set {vt : t¿k + 1} such that
a link between an exterior node associated with them
and the ith exterior node associated with vk exists in
G, and (iv) applying Theorem 1 if the condition holds.
As a consequence of the construction, the kth intermediate graph consists of the k rst stars of the grille
(with interior nodes v1 ; : : : ; vk and a set of exterior
nodes Wk ), a clique with p − k nodes (vk+1 ; : : : ; vp ),
a set of edges linking the exterior nodes of the stars
like in the grille, and more edges linking the exterior
nodes of the stars to the nodes in the clique as above.
Moreover, the facet which results from (4) is
k
X
((vj ) − 1)xj +
xj +
j∈Wk
j=1
6
X
k
X
j=1
((vj ) − 1) + 1:
p
X
j=k+1
xj
159
L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
Then, after p steps, inequality (3) should be reached
and the proof should be complete.
To end the proof, it is necessary to check that the
condition of Theorem 1 is satis ed in the kth step,
k = 1; : : : ; p. But note that Condition 2(d) of the Construction 2 guarantees that the ith exterior node associated with vk is either the only one linked to a star
or the only one linked to a node in the clique. If it is
a star, say the rst one, its exterior nodes along with
nodes v2 ; : : : ; vk−1 constitute the packing which is necessary to satisfy the condition. If it is a node, say vp ,
the packing includes vp plus the nodes v1 ; : : : ; vk−1 . In
any case, Theorem 1 can be applied and the proof is
complete.
4. New facets of the simple plant location polytope
Grilles are not an entelechy, but subgraphs of
well-known families of graphs can be identi ed as
grilles. In this section we obtain facets for a polytope
associated with the simple plant location problem,
based on grilles. For details about this problem, see
[6]. The interested reader will nd several families of
facets in [7] and Cornuejols and Thizy [9].
The simple plant location problem with p plants
and d destinations can be formulated as a set packing
problem with constraints
p
X
xij 61 ∀j ∈ {1; : : : ; d};
i=1
xij + yi 61 ∀i ∈ {1; : : : ; p}; ∀j ∈ {1; : : : ; d}:
We call G(p; d) the intersection graph associated
with this set of constraints, and label each node with
the name of its associated variable. Fig. 4 shows
G(4; 6).
Theorem 3. The subgraph of G(p; d) induced by a
set of nodes associated with a set of y-variables Qy
and a set of x-variables Qx such that
(i) yi ∈ Qy ∀i ∈ {1; : : : ; p};
(ii) ∀i1 ; i2 ∈ {1; : : : ; p} ∃j ∈ {1; : : : ; d} such that
xi1 j ; xi2 j ∈ Qx ;
(iii) ∀xij ∈ Qx ∃x‘j ∈ Qx ; ‘ 6= i; such that (xit ∈ Qx ⇒
x‘t ∈= Qx ∀t 6= j)
is a grille with set of interior nodes Qy and set of
exterior nodes Qx .
Fig. 4. Illustration of Example 3. All the edges of the complete
graph of 4 nodes have been separately considered.
Proof. It is clear that the subsets of nodes {yi }∪N (yi )
induce disjoint stars. Condition 2(a) of Construction
2 holds because two nodes xij1 ; xij2 are never linked,
2(b) follows from (ii) and (iii), 2(c) follows from (ii)
and 2(d) follows from (iii).
Corollary 1. Given G(p; d) and two sets of variables
Qx and Qy satisfying the conditions given in Theorem
3; the inequality
X
X
X
xij +
(
i − 1)yi 6
(
i − 1) + 1;
(5)
Qx
Qy
Qy
where
i := |{xij : xij ∈ Qx }|; is a facet of P(G(p′ ; d′ ))
for any p′ ¿p; d′ ¿d.
Proof. From Theorem 2, (5) is a facet of the subgraph
induced by Qx ∪ Qy . It is sucient to prove that any
other node in G(p′ ; d′ ) can be lifted with coecient
0 by means of Proposition 5.
Consider one of such nodes v, add it to a grille and
apply Proposition 5 (identifying v with vn ). There are
two possibilities:
1. v is not linked, in G(p′ ; d′ ), to any node of Qx ∪Qy ;
then, it is clear that n takes value 0.
2. v is a node xij with yi ∈ Qy ; then, v is linked
to yi and to the (maybe empty) clique of nodes
{xtj : xtj ∈ Qx }. In this case, a packing exists in
the grille that satis es (5) exactly and does not
contain any of the nodes linked to v. It is given by
160
L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
Fig. 5. Illustration of Example 4.
{y‘ : ‘ 6= i} ∪ {xit : xit ∈ Qx }. Therefore, n = 0 and
the proof is complete.
Example 3. The conditions of Theorem 3 indicate that, in order to construct a grille facet for
G(p; d), a complete graph of p∗ 6p nodes should be
edge-covered by a set of d∗ 6d complete subgraphs
in such a way that each node in each subgraph is the
end-point of an edge that is not in another subgraph.
In Fig. 4, the complete 4-graph has been decomposed
into single edges (thick lines). Consider the set
R = { (3; 1); (4; 1); (1; 2); (3; 2); (1; 3); (4; 3); (2; 4);
(3; 4); (2; 5); (4; 5); (1; 6); (2; 6) }:
The inequality
4
X
i=1
2yi +
X
xij 69
(6)
(i; j)∈R
is a facet of all the location polytopes with at least
4 plants and 6 destinations. Note that the y-nodes
(plants) have been numbered in the gure and the
x-cliques (destinations) have been sorted from left to
right.
Now, we show some fractional extreme points of
the polytope associated with the linear relaxation of
SPLP with p = 4 and d = 6 that are cut o by facet
(6). The rst point is given by
(
1=2 if (i; j) ∈ R;
2
1
1
1
1
y1 = y2 = y3 = y4 = 1=2; xij =
0 otherwise:
The second point is
y12 = 1;
y22 = y32 = y42 = 12 ;
2
2
2
2
2
2
2
2
2
= x41
= x32
= x43
= x24
= x34
= x25
= x45
= x26
x31
=
1
2
and xij2 = 0 otherwise. By symmetry, three more points
can be obtained from the latter. All of them are extreme points because the nodes associated with variables taking the value 1=2 induce connected subgraphs
containing an odd cycle, see [10]. It was also proved in
the cited paper that all the extreme points of the linear
relaxation of SPLP are cut o by clique and odd hole
inequalities. The following is an extreme point of the
polytope obtained when all the odd holes are added to
the linear relaxation of SPLP with p = 4 and d = 6:
(
1=3 if (i; j) ∈ R;
3
3
3
3
3
y1 = y2 = y3 = y4 = 2=3; xij =
0 otherwise:
The reader should check that this point is cut o by (6).
Of course, these points would be optimal solutions of
the linear relaxation of SPLP if the objective function
be similar enough to the left-hand side of the facet.
For instance, (x1 ; y1 ) would be the optimal solution
(maximum) if the objective function had the form
4
X
i=1
fi yi +
4 X
6
X
i=1 j=1
bij xij
L. Canovas et al. / Operations Research Letters 27 (2000) 153–161
with
fi = 200 ∀i;
bij =
(
100 if (i; j) ∈ R;
30 otherwise:
Example 4. In Fig. 5, a grille in G(7; 6) is shown
(thick lines), with associated facet
3y1 + 2y2 + y3 + 3y4 + 2y5 + 2y6 + y7
X
+
xij 615:
(7)
Qx
The black- lled node and the thin edges do not belong to the grille, but they illustrate the lifting process
of Corollary 1. The numbers out of the nodes are the
coecients of the nodes which constitute the packing
satisfying (7) exactly and do not belong to the neighborhood of the black- lled node.
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