www.elsevier.com/locate/dsw
New facets for the set packing polytope
Lazaro Canovas
a, Mercedes Landete
b, Alfredo Marn
a;∗aDepartamento de Estadstica e Investigacion Operativa, Facultad de Matematicas, Universidad de Murcia, Campus de Espinardo,
30071 Murcia, Spain
bCentro de Investigacion 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 plant location polytope. c 2000 Elsevier Science B.V. All rights reserved.
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. LetG= (V; E) be a graph with node set V and edge setE.G is said to beodd
(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
be-longs 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. Acomplete 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:amarin@um.es (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
cy-cle (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
e6∈ {e1; : : : ; e‘}linking two nodes in the cycle. Ahole
is a chordless cycle with more than three nodes. An
anti-hole is the edge-complement of a hole. A web
W(n; k), 16k6n=2, is a graph (V; E) such that|V|=n
andE=Sn
i=1{(i; i+k);(i; i+k+ 1); : : : ;(i; i+k+n)}.
Heren+ 1 is identied with 1 and so on. Ananti-web
is the edge-complement of a web. Theneighborhood
N(v) of a nodevis the set of nodes that are adjacent to v. The incidence degree (v) of a node v is the cardinality of its neighborhood. Astaris 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)
associated withG,P(G), is the convex hull ofPI(G).
(It holds thatP(G) is a full-dimensional polytope, and a vectorxis a vertex ofP(G) if and only ifx∈PI(G)).
A set packing problem is a binary optimization problem
SPP : Opt{cx:Ax61m; x∈ {0;1}n};
wherec∈Rn,A∈ {0;1}m×nand1
mis anm-vector of
ones. The graph associated with(intersection graph of) SPP isG= (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 isPI(G) and the optimal
solutions of SPP can be obtained by solving the linear optimization problem Apart from thetrivialfacets−xj60, all the facets of
P(G) verifyj¿0∀jand0¿0 (nontrivialfacets).
Up to multiplication by a positive constant, there is a unique set of facetsix6i
0; i= 1; : : : ; ‘, such that P(G) ={x:ix6i
0; i= 1; : : : ; ‘}. A set of linear
in-equalities satisfying the last condition is called a den-ing linear system of P(G). Since set packing prob-lems 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 inter-est to contribute to the characterization of the dening linear system ofP(G), i.e., to obtain facets ofP(G).
Sets of nodes will be usually denoted byV or Vi,
and the same node will be denoted indierently byvj
andj. In particular,jwill be used in the gures, sum-mations and subindices andvjin the text. Frequently,
the expressionfacet of the graphwill be used instead offacet 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 associ-ated known facets, and to devise methods for trans-forming 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 dening graphs: Cliques, odd holes, webs, odd anti-holes and anti-webs.
Proposition 1 (Nemhauser and Trotter Jr. [10] and Padberg [11,13]).LetG= (V; E)be a graph and let B be a subset of V. The inequalityP
j∈Bxj61is a
facet ofP(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 calledclique facet.
Proposition 2 (Nemhauser and Trotter Jr. [10] and Padberg [11]).LetG= (V; E)be an odd hole. Then;
the inequality P
j∈Vxj6(|V| −1)=2 is a facet of
P(G).
Proposition 3 (Nemhauser and Trotter Jr. [10] and Padberg [11]). Let G= (V; E) be an odd anti-hole. Then;the inequalityP
j∈V xj62is a facet ofP(G).
Proposition 4( Padberg [13] and Trotter [14]). Let
G=W(n; k)be a web. Then;
1. The inequalityPn
j=1xj6k is a facet of P(G) if
and only ifg:c:d:(n; k) = 1.
2. If g:c:d:(n; k) = 1 and k ¿1; the inequality
Pn
j=1xj6⌊n=k⌋is a facet ofP( G).
A method for obtaining facets ofP(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.
More transformations of pairs graph-facet can be consulted in [13], Cho et al. [7] and Wolsey [15]. Ad-ditional families of graphs and associated facets have been studied in the literature, see [2,5] for wheels, see [2,3] forseries–paralleland other special graphs. In Canovas et al. [4], new results concerning the fa-cial structure of set packing polyhedra are presented; in particular, new facet generating methods and facet dening 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 lift-ing several coecients. A dierent approach is to consider facets associated with graphs of lower di-mension, 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 dierent generalization. It is important to notice that we start with the “small” graphG and with a facet of P(G) and give the method to obtain the “large” graph, which we call Gg; and a facet
ofP(Gg):
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 Gg is
Fig. 1. Illustration of Construction 1.
obtained by
1. Covering the neighborhood ofvn,N(vn), by means
ofmsets of nodesVi⊂V,Vi6=∅; i= 1; : : : ; m(i.e.
N(vn) =Smi=1Vi).
2. Introducing m new nodes vn+i and linking each
vertex inVitovn+i.
3. Linkingvntovn+i,i= 1; : : : ; m, only.
Note that the subsetsViare not necessarily pairwise
disjoint. Wolsey’s transformation can be obtained as a particular case whenm= 2 andV1∩V2=∅, and then
Proposition 3 in [15] is obtained from the following result.
Theorem 1. LetPn
j=1jxj60be a nontrivial facet
ofP(G)and dene
(Q) := max
n X
j=1
jxj:x∈PI(G); x‘= 0
∀v‘∈
[
t∈Q
Vt
∪ {vn}
forQ⊂{1; : : : ; m}.If (Q) =0 for all Q such that |Q|=m−1;then the inequality
n−1
X
j=1
jxj+ (m−1)nxn+ m X
i=1 nxn+i
60+ (m−1)n (1)
Proof. First, it will be shown that (1) is a valid in-equality forP(Gg) and then thatn+manely
indepen-dent points inPI(Gg) exist satisfying (1) exactly. Note
that, from the assumptions of the theorem, (Q) =0
dened pointx2 it holds
0¿ ones with a zero in the ith position, are vertices of
P(Gg) and satisfy (1) exactly. Moreover, then+m−1
rst points are clearly a set of anely independent points ofRn+m, and assuming
Example 1. LetG be the left-hand graph of Fig. 2
vn=v15. Setm= 4,V1={1;2},V2={2;3},V3={4}
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 satises (2) exactly
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} andV2∪V3∪V4∪ {v15},
re-spectively. Therefore, the following facet ofP(Gg) is
obtained:
8
X
j=1
xj+ 3x9+x10+x11+ 2x12+x13+x14+ 3x15
+x16+x17+x18+x19610:
3. Grilles
This section is devoted to the construction of a new family of facet dening 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 calledexterior
nodes associated with an interior node (the neigh-bor), 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 nodeve associated with the
interior node vi, another interior node vj,j∈
{1; : : : ; p}−{i}exists so thatN(ve)∩N(vj)6=
∅andN(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 satises 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 exte-rior node indicates a possible value forjin condition 2(d) of the Construction 2.
Theorem 2. The inequality
p X
j=1
((vj)−1)xj+ n X
j=p+1 xj6
p X
j=1
((vj)−1) + 1
(3)
denes 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 graphQwith nodes
{v1; : : : ; vp},p times, (i) Q becomes G and (ii) the
facet Pp
j=1xj61 ofP(Q) results in the facet (3) of
P(G).
LetQ be the 0th intermediate graph. Assume that the (k−1)th intermediate graph,k= 1; : : : ; p, consists of the rstk−1 stars of the grille (with interior nodes
v1; : : : ; vk−1and a set of exterior nodesWk−1), a clique
Fig. 3. Illustration of Example 2. Black-lled nodes areinteriornodes, 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 nodevto any exterior node of the star associated with the interior node vh. Moreover,
assume that
k−1
X
j=1
((vj)−1)xj+ X
j∈Wk−1
xj+ p X
j=k
xj
6 k−1
X
j=1
((vj)−1) + 1 (4)
is a facet of this intermediate graph. It can be easily checked thatQsatises 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
se-lected node, (ii) settingmequal to the number of ex-terior nodes of thekth star in the grille, (iii) choosing
Vi,i= 1; : : : ; m, equal to the set of nodes that are either
(a) exterior nodes associated withv1; : : : ; vk−1 which
are linked, in G, to the ith exterior node associated withvkor (b) nodes of the set{vt:t¿k+ 1}such that
a link between an exterior node associated with them and theith 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 inter-mediate graph consists of thekrst stars of the grille (with interior nodes v1; : : : ; vk and a set of exterior
nodesWk), a clique withp−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
j=1
((vj)−1)xj+ X
j∈Wk
xj+ p X
j=k+1 xj
6 k X
j=1
Then, afterpsteps, 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 satised in the kth step,
k= 1; : : : ; p. But note that Condition 2(d) of the Con-struction 2 guarantees that theith exterior node asso-ciated 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 nodesv2; : : : ; vk−1constitute the packing which is
nec-essary to satisfy the condition. If it is a node, sayvp,
the packing includesvpplus the nodesv1; : : : ; 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 identied 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 andddestinations can be formulated as a set packing problem with constraints
p X
i=1
xij61 ∀j∈ {1; : : : ; d};
xij+yi61 ∀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 ofx-variablesQx such that
(i) yi∈Qy ∀i∈ {1; : : : ; p};
(ii) ∀i1; i2∈ {1; : : : ; p} ∃j∈ {1; : : : ; d} such that xi1j; xi2j∈Qx;
(iii) ∀xij∈Qx∃x‘j∈Qx; ‘6=i;such that(xit∈Qx⇒
x‘t∈=Qx ∀t6=j)
is a grille with set of interior nodes Qy and set of
exterior nodesQx.
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 nodesxij1; xij2 are never linked,
2(b) follows from (ii) and (iii), 2(c) follows from (ii) and 2(d) follows from (iii).
Corollary 1. GivenG(p; d)and two sets of variables
QxandQysatisfying the conditions given in Theorem
3;the inequality
X
Qx
xij+ X
Qy
(i−1)yi6 X
Qy
(i−1) + 1; (5)
wherei:=|{xij:xij∈Qx}|;is a facet ofP(G(p′; d′))
for anyp′¿p; d′¿d.
Proof. From Theorem 2, (5) is a facet of the subgraph induced byQx∪Qy. It is sucient to prove that any
other node inG(p′; d′) can be lifted with coecient 0 by means of Proposition 5.
Consider one of such nodesv, add it to a grille and apply Proposition 5 (identifyingvwithvn). There are
two possibilities:
1. vis not linked, inG(p′; d′), to any node ofQ x∪Qy;
then, it is clear thatn takes value 0.
2. v is a node xij with yi ∈ Qy; then, v is linked
toyi and to the (maybe empty) clique of nodes
{xtj:xtj ∈ Qx}. In this case, a packing exists in
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 indi-cate that, in order to construct a grille facet for
G(p; d), a complete graph ofp∗6pnodes should be edge-covered by a set of d∗6dcomplete 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
(i; j)∈R
xij69 (6)
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 withp= 4 andd= 6 that are cut o by facet (6). The rst point is given by
y21=y12=y31=y41= 1=2; x1ij=
(
1=2 if (i; j)∈R;
0 otherwise:
The second point is
y21= 1; y22=y23=y42=12;
x231=x412 =x322 =x432 =x242 =x234=x225=x245=x226
=12
andx2
ij= 0 otherwise. By symmetry, three more points
can be obtained from the latter. All of them are ex-treme points because the nodes associated with vari-ables 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 withp= 4 andd= 6:
y31=y32=y33=y34= 2=3; xij3 =
(
1=3 if (i; j)∈R;
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 fiyi+
4
X
i=1 6
X
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
Qx
xij615: (7)
The black-lled node and the thin edges do not be-long 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 neigh-borhood of the black-lled node.
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