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Chapter 19. A ’Pseudopolynomial’ Algorithm for Sequencing Jobs to Minimize Total Tardiness

Chapter 19. A ’Pseudopolynomial’ Algorithm for Sequencing Jobs to Minimize Total Tardiness

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The minimal integral separator


Proof. By (2) and property P for any non-stable set S,, x l E r a 2

i b + 1. If S, is stable

then either I C {1,2,3,. . . , t } or else I C {1,2, . . . , sj - 1,j } for some j. In either case

by (1) and (3) we must have that xCi,,aiG b.

Acknowledgement. Partial support through NRC (Grant A 8552) is gratefully



[l] V. Chvital and P.L. Hammer, Aggregation of Inequalities in Integer Programming, Ann. Discrete

Math. 1 (1977) 145-162.

This Page Intentionally Left Blank

Annals of Discrete Mathematics 1 (1977) 421-434

Publishing Company

0 North-Holland


Manfred W. PADBERG

Graduate School, Faculty of Business Administration, New York University, New York, N Y 10006,


We review some of the more recent results concerning the facial structure of set packing

polyhedra. Utilizing the concept of a facet-producing graph we give a method that can be used

repeatedly to construct (arbitrarily) complex facet-producing graphs. A second method, edgedivision, is used to further enlarge the class of facet-defining subgraphs.

1. Introduction

We consider the set packing problem (SP)

max ck


x i = O or 1 f o r j = 1 , ..., n

where A is a m X n matrix of zeros and ones, e T = (1,. . ., 1) is a vector of m ones

and c is a vector of n (arbitrary) rational components. This class of combinatorial

optimization problems has recently received much attention, both as a problem of

considerable practical interest -see e.g. [3,10,22] for recent survey articles for this

problem and its close relatives - as well as a combinatorial programming problem

that captures most of the difficulties and computational complexities that are

present in the general zero-one programming problem [2, 10, 11, 151.

In this paper we extend the class of “strongest” cutting planes or facets known

for problem (SP) and show that for set packing problems of sufficiently large size

arbitrarily “complex” valid inequalities can be constructed that, however, are facets

of the convex hull of (integer) sohtions to (SP), i.e. belong to the class of linear

inequalities that uniquely define the convex hull of solutions to (SP). Without

restriction of generality, we will assume throughout the paper that A does not have

any zero column or zero row. Denote by P = P(A, e ) the polyhedron given by the

feasible set of the linear programming problem associated with (SP), i.e.

P(A,e ) = {x € R“ I Ax


e, x



* Parts of this paper were presented in preliminary form at the NATO Advanced Study Institute

Symposium on Cornbinatorial Programming held in Versailles, France, September 1974.

42 1

M.W. Padberg


Furthermore, let PI = P l ( A ,e ) denote the set packing polyhedron, i.e. the convex

hull of integer points of P

~ , ( ~ , e ) = c o n v {ExP ( A , ~ ) integer}.


By the theorem of Weyl[25], there exists a finite system of linear inequalities whose

solution set coincides with PI, i.e.

PI = { x E R " ( H x

h,x S O }


for some appropriate matrix H and vector h. Some research activity has recently

focused on identifying part (or all) of those linear inequalities that define PI, see [5,

16, 17, 18,20,21,23,24]. This interest is motivated in part by the desire to use linear

programming duality in proving optimality - with respect to the linear form cx of a given extreme point of PI.As every extreme point of PI is also an extreme point

of P and hence of any polyhedron P satisfying PI C P P, it is generally sufficient

to work with a partial - rather than a complete - linear characterization of PI.

More precisely, one is interested in finding part (or all) of the linear inequalities that

define facets of PI. Note that dim P = dimPI = n, i.e. both P and PI are fully

dimensional. As customary in the literature, we will call an inequality n-x G r0 a

facet of PI if (i) T X s m0 for all x E PI and (ii) there exist n affinely independent

vertices x ' of PI such that n-x' = nofor i = 1 , . . ., n. One readily verifies that each

inequality x, 2 0, j E N, is a (trivial) facet of PI, where N = {I,. . ., a } .

A construction that has proved useful in identifying facets of PI is the

intersection graph associated with the zero-one matrix A defining P. Denote by a,

the j t h column of the m x n matrix A . The intersection graph G = ( N ,E ) of A has

one node for every column of A, and one (undirected) edge for every pair of

nonorthogonal columns of A, i.e. (i, j ) E E iff a,u, 2 1. One verifies readily that the

weighted node packing problem (NP) on G for which the node weights equal c, for

j = 1 , . . ., n is equivalent to (SP), i.e. (NP) has the same solution set and set of

optimal solutions as the problem (SP). (The weighted node packing problem (NP)

on a finite, undirected, loopless graph G is the problem of finding a subset of

mutually non-adjacent vertices of G such that the total weight of the selected

subset is maximal, See [6, 10, 15, 181 for more detail on the stated equivalence.)

This observation is very useful as it permits one to restrict attention to node packing

problems in certain subgraphs of G when one tries to identify facets of PI.

2. Facet producing subgraphs of intersection graphs

Let n-x T,, be a non-trivial facet of PI. We can assume without loss of generality

that both rrJ,j E N, and r 0are integers. From the non-negativity of A, it follows

readily that 71; 2 0 for all j E N and .rr0>0. For suppose that N - = { j E N

T, < 0) # 0 for some non-trivial facet r x s noof PI. As r x s r 0is generated by n

affinely independent points of PI, there exists an X E PI such that TX = no and



Complexity of set packing polyhedra

2, = 1 for some j E N - . (For, if not, then by the assumed affine independence we

have that I N-(

= 1. It follows that r o= 0 and since all unit vectors of R" are

- feasible

that r x < r(,is a triuial facet of the form x, 2 0 ) . But the point given by


x, = Z,, j_E N - N-, = 0, j E N - is contained in PI as A is non-negative and

hence, r: > r owhich is impossible. Consequently, every non-trivial facet r x s ro

satisfies r,2 0 for all j E N and r o > O .

Denote by S = S ( r ) the support of r , i.e. S = { j E N r,> O}. Let Gs = (S, E s )

be the induced subgraph of G with node set S and edge set Es C E and let

P s = P n { x E R" x, = 0 for all if?!!

S } . D u e to the non-negativity of A, one verifies

readily that the convex hull PS of integer points of P s satisfies P S =

PI n {x E R" x, = 0 for all j E S } . Furthermore, n-x S r oretains its property of

being a facet of PS. If one considers proper subgraphs of Gs, this property of

r x s r o may or may not be "inherited." In fact, if all components of T are

(non-negative) integers and r o= 1, then one readily verifies that for all subgraphs

(including those on single nodes) the property of r x rot o be a facet of the

resulting (lower-dimensional) packing polyhedron is retained. The next theorem

characterizes all facets of P, with integer r and r o =1, see [9, 181.





Theorem 1. The inequality c,,,x, S 1, where K C N,is a facet of PI if and only i f

K is the node set of a clique (maximal complete subgraph) of G.

Thus one knows all subgraphs of the intersection graph G of a zero-one matrix

A that give rise to facets r x rowith nonnegative integer r,,j = 1,. . ., n, and

r o= 1. If, however, r oa 2 and r odoes not divide all components of r,then there


exists a smallest subgraph G' of G, G' not an isolated node, such that T X

looses its property of being a facet of the packing polytope associated with the

packing problem of any proper subgraph of G'. If r >O, i.e. if S = N, then, of

course, the (full) intersection graph G may have this property and thus G may be

itself strongly facet-producing [24].

Definition. A vertex-induced subgraph Gs = ( S , E s ) of G = (N, E ) with node set

S C N is facet-producing if there exists an inequality r x s 7r0 with nonnegative

integer components r, such that (i) r x s rois a facet of PS = PI n { x E R" x, = 0

for all j$Z S } and (ii) r x S m0 is not a facet for PT= PI n {x E R" x, = 0 for all

j e T } where T is any subset of S such that I T 1 = IS I - 1. A subgraph Gs of G is

called strongly fucet-producing if there exists an inequality r x c rosuch that (i)

holds and (ii) holds for all T C S satisfying I T 1 G 1 S 1 - 1. A subgraph Gs of G is

facet-defining if there exists an inequality r x s r,,such that (i) holds and (iii) such

that r, > O for j E S. (Shortly we will say that Gs defines the facet r x s ro.)



Remark 1. Every strongly facet-producing (sub-)graph is facet-producing. Every

facet-producing (sub-)graph is facet-defining. If Gs is facet-defining, then Gs is


M. W . Padberg


Proof. The first two parts being obvious, let rrx c rro be a facet defined by

Gs = ( S , Es), i.e. rr, > O for j E S, and suppose that Gs is not connected. Then

G, = ( S , E , ) can be written as Gs = G I U G, where G, = ( S , , E , )with S,# 0 for

i = 1 , 2 and S = S , U S,, S , f l Sz = 0 and Es = E l U EZ.Let P i be defined like P s

with S replaced by S,, and let rri = max {rrx x E P i } for i = 1,2. Since T X c


defined by Gs, it follows that rr; > 0 for i = 1,2. Define TTT; = rr, for j E S,, rrf = 0

for j e S,, and let x ’ E P ; be such that rrx‘ = a i for i = 1 ,2 . Since Gs is

disconnected, x ’ + x z E P: and hence, rrX + rri c T o . It follows that every x E PS

satisfying nx = rro satisfies rr’x = IT,!, for i = 1 , 2 and consequently, rrx =G

is not a

facet of PS.

By the discussion preceeding the definition, it is clear that every facet rrx s T o of

P, is either produced by the subgraph G, having S = S(T) or if not, that there

exists a subgraph GT of Gs with T C S such that the facet i i x =G m0 of PT is

produced by GT where 71, = rrl for j E T, i;, = 0 for jE T and PT is defined as

previously. The question is, of course, given i i x < no can we retrieve the facet

rrx n o of PI. The answer is positive and follows easily from the following theorem

which can be found in [16].



Theorem 2. Let P s = PI fl { x E R” x, = 0 for all j @ S} be the set packing polyhedron obtained from PI by setting all variables x,, j E N - S, equal to zero. If the


a,xl s a. is a facet of P?, then there exist integers &, 0 /?, ao,such

l is a facet of PI.

that Z J E S q x+I Z , E N - s ~ Jsx a.


Generalizations of Theorem 2 for more general polyhedra encountered in

zero-one programming problems have been discussed in [l, 4,12,14,19,26,28].

The apparent conclusion from this result - in view of the notion of the

intersection graph discussed above - is that the problem of identifying part (or all)

of the facets of a set packing polyhedron P, is thus equivalent to the problem of

identifying all those subgraphs of the intersection graph G associated with a given

zero-one matrix A that are facet-producing in the sense defined above. (It should

be noted that the “facet-defining’’ property of (sub-)graphs is a considerable weaker

property as in this case we require solely that the corresponding facet has positive

coefficients. The choice of terminology may seem somewhat arbitrary, but the

positivity of all components of a facet furnished by a facet-defining (sub-)graph is

crucial in some of the arguments to follow.) O n e possible attack o n the problem of

finding a linear characterization of PI is thus t o “enumerate” all possible graphs

that are facet-producing, a truly difficult task as we will show in the next section.

3. Facet-producing graphs

Let G = (N, E ) be any finite undirected graph having no loops. Denote by Ac

the incidence matrix of all cliques of G (rows of A,) versus the nodes of G

Complexity of set packing polyhedra



(columns of Ac). Let N = (1,. . ., n } and let Pc = {x E R“ Acx S ec, x a 0) where

ec = (1,. . ., 1) is dimensioned compatibly with Ac. As before let PI denote the

convex hull of integer points of Pc,i.e. PI = conv{x E Pc x integer). We will not

note explicitly the dependence of t h e respective polyhedra upon the graph G which

may be taken as the intersection graph associated with some given zero-one matrix.

The term of a facet-producing (facet-defining) graph is used here analogously with

S = N. If x is a node (edge) of G, then by G - {x} we will denote the graph

obtained from G by deleting node x from G and all edges incident to x from G (by

deleting the edge x, but no node from G). Denote by G the complement of G, i.e.

G = (N, % ( N )- E ) where % ( N ) is the set of all edges on n nodes. Every clique in

G defines a stable (independent) node set (or node packing) in G and every

maximal stable node set in G defines a clique in G. Let Qc = {x E R“ Bcx s

dc, x 3 0) where Bc is the incidence matrix of all cliques in G and d z = (1,. . ., 1) is

dimensioned compatibly. Furthermore, let QI = conv{x E Qc x integer}. One

verifies readily that Qc is the anti-blocker of PI and that Pc is the anti-blocker of

QI, see [9, 201.

Before investigating special facet-producing graphs it is interesting to note the

following proposition which substantially reduces the search for facet-producing

graphs. Contrary to what one might expect intuitively, it i s not necessarily true that

the complement C? of a facet-producing graph G is again facet-producing. The

graph in Fig. 1 shows an example of a facet-producing graph G whose complement

G is not facet-producing. In fact, whereas the graph G of Fig. 1 produces the facet

x, s 4, its complement C? does not produce any facet in the sense of the above

definition; rather, every facet of the associated packing problem is obtained by

“lifting” the facets produced by some proper subgraph of G, as the clique-matrix of

G is of rank 9.





Fig. 1.

To state the next theorem we meet the following definition from linear algebra:

A square matrix M is said to be reducible if there exist permutation matrices P and

Q such that


R ]


where N and R are square matrices and 0 is a zero-matrix. If no such permutation

matrices exist, then M is called irreducible.

Theorem 3.

Suppose that G defines the facet


s r0 for PI such that


M . W . Padberg


max { r r x x E Pc}= r r f is assumed at a vertex X of Pc satisfying 0 < 2, < 1 for

j = 1,.. ., n. If the submatrix A 1of Acfor which AIX = e, is irreducible (and square),

then the complement graph

G is

strongly facet-producing.

Proof. Since every row of Ac defines a vertex of Q I , it follows from the

assumption that O < X j < 1 for all j = 1, .. ., n that the hyperplane X y = 1 is

generated by n linearly independent vertices of Q I . On the other hand, AcX ec

implies validity of X y S 1 for Q17 i.e. QI { y E R” Xy S l}.Hence, G defines the

facet Xy s 1 of Q,. Suppose that Xy 6 1 defines a facet for some k-dimensional

polyhedron Q, C QI satisfying k < n. Then there exist k linearly independent

vertices y ’ of 6,satisfying X y ‘ = 1. Since the submatrix A , of A, defining X is

square, it follows that upon appropriate reordering of the rows and columns of A 1 ,

A , can be brought into the form (3.1) contradicting the assumed irreducibility

of A , .

As an immediate consequence we have the corollary:



Corollary 3.1. If G defines a facet rrx rro of PI such that max{n-x x E Pc} = rrf

is assumed at a vertex X of Pc satisfying 0 < 2, < 1 for all j = 1,. . ., n, then defines

a facet of QI.

Chordless odd cycles (“holes”) as well as their complements (“anti-holes”) are

known to define facets. In the former case one readily verifies the hypothesis of

Theorem 3 to conclude that anti-holes as well as holes are strongly facet-producing.

More recently, L. Trotter [24] has introduced the notion of a “web” which properly

subsumes the aforementioned cases: A web, denoted W ( n ,k ) is a graph G = (N,


such that

IN/= n 3 2 and for all i, j E N, ( i 7 j ) EE iff j =

i + k , i + k + 1,.. ., i + n - k , (where sums are taken modulo n ) , with 1 S k S [ n / 2 ] .

The web W ( n ,k ) is regular of degree n - 2k + 1, and has exactly n maximum node

packings of size k . The complement W ( n , k ) of a web W ( n ,k ) is regular of degree

2 ( k - 1 ) and has exactly n maximum cliques of size k . O n e verifies that W ( n ,1) is a

clique o n n nodes, and for integer s 3 2 , W ( 2 s + 1, s ) is an odd hole, while

W ( 2 s + 1 , 2 ) is an odd anti-hole. The following theorem is essentially from [ 2 4 ] ,see

the appendix.

Theorem 4. A web W ( n ,k ) strongly produces the facet C,”=,x, k i f and only if

k 2 2 and n and k are relatively prime. The complement W ( n , k ) of a facetproducing web W ( n ,k ) defines (strongly produces) the facet

x, s [ n / k ] ( i f and

only i f n = k [ n l k ] + 1).


The next theorem due to V. Chvital [5] provides some graph-theoretical insights

into graphs that give rise to facets with zero-one coefficients. To this end, recall that

an edge e of a graph is called a-critical if a ( G - e ) = a ( G ) +1, where a ( G )

denotes the stability number of G, i.e. the maximum number of independent nodes

of G.

Complexity of set packing polyhedra


Theorem 5. Let G = ( V ,E ) be a graph ; let E * E be the set of its (Y -critical edges.

If G * = ( V , E * ) is connected, then G defines the facet &,x,


It would be interesting to know whether all facets of set packing polyhedra

having zero-one coefficients and a positive right-hand side constant can be

described this way. (The question has been answered in the negative by Balas and

Zemel [4a]). V. Chvatal also discusses in his paper [5] several graph-theoretical

operations (such as the separation, join and sum of graphs) in terms of their

polyhedral counterparts. Though very interesting in their own right, we will not

review those results here. In particular, the two constructions given below are not

subsumed by the graphical constructions considered by V. Chvbtal.

We note next that graphs that satisfy the hypothesis of Theorem 5 need not be

facet-producing in the sense defined in Section 2. In fact, the graph of Fig. 2

provides a point in-case. The facet defined by the graph G of Fig. 2 is given by

x, 2 which, however, is produced by the odd cycle on nodes {1,2,3,4,5}. The

x, S 2, i.e. by applying

coefficient of x6 is obtained by “lifting” the facet

Theorem 2 .





Fig. 2

W e next turn to a construction which permits one t o “build” arbitrarily complex

facet-producing graphs. Let G be any facet-defining graph with node set V =

(1,. . ., n } with n 3 2 and consider the graph G * obtained by joining the ith node of

G t o the ith node of the “claw” K1,”by an edge. The claw K,.,,- also referred to

as a “cherry” or “star”, see [13] - is the bipartite graph in which a single node is

joined by n edges to n mutually non-adjacent nodes. W e will give the node of K , , ,

that is joined to the ith node of G the number n + i for i = 1,. . ., n, whereas the

single node of Kl,,, that is not joined to any node of G, will be numbered 2n + 1.

(See Fig. 3 where the construction is carried out for a clique G = K4.) Denote by

V * = (1,. . ., 2n + 1) the node set of G * and by E * its edge-set. It turns out that G *

is facet-defining (this observation was also made by L. Woolsey [27]), and

moreover, that G * is strongly facet-producing.

Fig. 3.

Theorem 6. Let G = ( V , E ) be a graph on n 2 2 nodes and let T X S no be a

(non-trivial)facet defined by G. Denote by G * = ( V * ,E*)the graph obtained from

M . W. Padberg


G by joining every node of G to the pending notes of the claw K1,"a s indicated above.

Then G * strongly produces the facet

where x ( ' ) = ( x l , . . ., x,,), x(')= ( x " + ~.,. ., x2") and x ~ are

~ the

+ ~variables of the

node-packing problem on G * in the numbering defined above.

Proof. Let A, be the clique-matrix of G and denote by PI the set packing

polyhedron defined with respect to A,. The clique matrix of G * is given by A $:

A, 0 0

A$='[ I I 01

0 I e


where I is the n x n identity matrix, e is vector with n components equal to one,

and 0 are zero-matrices of appropriate dimension. Denote by PT the set packing

polyhedron defined with respect to A 2.. To establish validity of (3.2)for P : we note

that x, = 1 for some j E { n 1, , . ., 2 n ) implies that x ~ , , +=~0. Consequently, as

+ x(2) < e, every vertex of PT having x, = 1 for some j E { n + 1,. . ., 2 n ) satisfies

(3.2). On the other hand, since r x ( l ) < nofor every vertex of PT and no<

n;, it

follows that every vertex of PT satisfies (3.2). To establish that the inequality (3.2)

defines a facet of PT, let B denote any n X n nonsingular matrix whose rows

correspond to vertices of PI satisfying T X < n,, with equality. Then define matrix

B * as follows:






where Z is the n x n identity matrix, E is the n X n matrix with all entries equal to

one, e is the vector with n components equal to one, and 0 are zero-matrices of

appropriate dimension. O n e verifies that the absolute value of the determinant of

B* is given by


Hence, B * is non-singular since det Bf 0 and

r, > r,,> 0. On the other hand,

every row of B * corresponds to some vertex of PT satisfying (3.2) with equality.

Consequently, (3.2) defines a facet of PT. To prove that G * produces a facet in the

sense defined above, we show that no graph G * - { j } defines the facet (3.2) for

j = 1 , . . ., 2n + 1. Note first that from the positivity of rr it follows that every vertex

of PT satisfying (3.2) with equality satisfies x, + xn+, + x z n + ,3 1 for all j E V. Let

now j E V and consider G * - { j } . Every vertex of PT satisfying x, = 0 and (3.2)with

equality, necessarily satisfies x,+, + x ~ , , +=~1. Consequently, (3.2) does not define a

42 9

Complexity of set packing polyhedra

facet of P, = P: n{x E R * " + ' I x ,= 0) for j E V. Consider next vertices of P t

satisfying (3.2) with equality and x,+, = 0 for j E V. Since n 3 2 and rrx rro is

defined by G, it follows that the node j of G has a neighbor k ( j ) in G, i.e.

(j, k (j))E E for some k (j) # j , k (j) E V. Consequently, every vertex satisfying (3.2)

with equality and x,+, = 0, also satisfies the equation x n+k(,)+ x ~ = ~1. Hence

+ ~ (3.2)

does not define a facet of Pa+, for j E V. Finally, every vertex of P t satisfying (3.2)

with equality and xZ,,+'= 0 satisfies the equation x k x " + ~= 1 for all k E V, and

Consider next a subgraph

consequently, (3.2) does not define a facet of Pzn+l.

G' = (V', E ' ) of G * having I V'I s 2n - 1 nodes. If G' defines the facet given by

(3.2), then, as noted earlier, all vertices of P' = PT n { x E R2"+'x, =

0, j E V* - V'} satisfying ( 3 . 2 )with equality must satisfy x, + xn+, + x ~ 3 ~1 for+ all~

j E V, since a vertex of P' is also a vertex of P : . It follows that V' must contain the

V' implies n + i E V' for all

node numbered 2n + 1 and furthermore, that

i E V. Consequently, V C V' and V' n { n + 1 , . . ., 2 n } # 0. Let N ' C { n f 1 , . . ., 2 n }

be the nodes of G * that are not in G'. Then either there exists a node n + j E N'

such that the node j E V has a neighbor k (j) E V satisfying n + k (j)E V' or else,

G is disconnected. The latter contradicts Remark 1. Consequently, by the above

reasoning, we have N ' = 0, i.e. V = V'. This completes the proof of Theorem 6.




Corollary 6.1. Let G, G * and rr be a s in Theorem 6 . If there exists a vertex X E Pc

such that max {rrx x E Pc}= rr2 is assumed at vertex 2 of Pc satisfying 0 < X, < 1

for j = 1, . . ., n, then the complement graph G * of G * defines a facet. Moreover, this

facet of the set packing polyhedron Q T associated with G * ( i n rational form ) is given


. + ( e - 2 ) . x ( * ) + f .x Z n i l s 1



where f



min {X, j = 1,. . ., n } .

Proof. Using the clique-matrix A T.as defined by (3.3) o n e verifies readily that the

coefficients of the inequality (3.5) define a vertex of PT.with all components strictly

between zero and one. Furthermore, the submatrix of A T.defining the vertex with

components (X,e - X, Z) is nonsingular. As the cliques in G *define vertices of 0 T,

Corollary 6.1 follows.

The second construction uses edge-division. Let G b e any facet-defining graph

with node set V = (1,. . ., n } and edge-set E. Let e = ( v , w ) E E and consider the

graph G * with nodes set V* = (1,. . ., n, n + 1 , n + 2) and edge-set

E * = ( E - { e } )U {(u, n

+ l),( n + 1 , n + 2), ( n + 2 , w ) } .

That is, G * is obtained from G by "inserting" two new nodes into an (existing)

edge of G. Let rrx rro be the facet defined by G. As usual, we will assume that rr is

a vector of positive integers. An edge e = (u, w ) E E will b e called rr-critical if

there exists an independent node set F in the graph G - { e } such that x , € F q > no


rr, = rro or

rr, = rro. Note that rr-criticality of an edge is entirely

analogous to the concept of (Y -criticality used above.



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