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Chapter 20. Complexity of Machine Scheduling Problems

# Chapter 20. Complexity of Machine Scheduling Problems

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43 1

Complexity of set packing polyhedra

c;=I+

T X =s T,, produced by the graph is given by

x, 3x9 4. As o n e readily verifies,

every edge of G is r-critical. Consequently, we can insert into any o n e of the edges

of G two nodes; taking e = (8,9) we get a new graph G * and associated facet is

x, + 3 x y + x l o + xll s 5. Anyone of the edges of the graph G * is again .rr-critical

and we can continue inserting pairs of nodes into its edges, etc. Returning to the

graph of Fig. 3 and adding a node 10 that is joined by edges t o nodes 5 , 6 , 7 , 8 and 9,

we get from Theorem 2 the following facet defined (not produced) by the enlarged

graph G’: %=,x, + 3xy+ 3x10 < 4. Upon inspection, we find that the (9,lO) of G‘ is

r-critical. Inserting two nodes in the way described in Theorem 7 we obtain the

facet defined by the resulting graph to be given by

x, + 3x9+ 3x10 + 3x11 +

3xI2 < 7. Using the construction of Theorem 6, we can get a fairly complex looking

facet.

O n e might suspect from the foregoing that, given any set of positive integers

do,d , , .... d, satisfying d, < do for j = 1,. ... n, at least four d, = 1 and

d, > do,

there exists a graph G producing a facet r x s r0 such that .rr, = d, for j =

0,1,. ... n, prorided that n is chosen sufficiently large. (The answer to this problem

is definitely in the negative for small n.) My guess is that the answer is positive.

The foregoing may suggest that the complexity of the facial structure of set

packing polyhedra renders useless pursuit of this line of research as regards its use

in any computation utilizing linear programming relaxations. T h e following example may serve to indicate the contrary and points to an interesting question that,

presumably, can only be answered in a statistical sense.

cy=,

cy,l

c,”_l

Example. Consider the maximum-cardinality node-packing problem on an odd

anti-hole G with n 2 5 vertices and let AG denote the edge vs. node incidence

matrix of G. Denote by R the following permutation matrix:

-0

1 0

.....0

0 0 1 0...0

R=

i

. . . . . . . . .0 1

-1 0 . . . . . . . . 0

0

We can write A : = (AT, .... A 3 where p = [ n / 2 ] - 1 and A T = ( I + R’)=for

i = 1,. . ., p with Z being the n x n identity matrix. Let P = {x E R” AGx s e,

x 2 0 } be the linear programming relaxation of the node-packing problem and PI

the convex hull of integer solutions. As one readily verifies, max{c:;=, x, x E P } =

n / 2 for all n. But, the integer answer is two, n o matter what value n assumes, i.e.

rnax{C;=, x, x E P l } = 2 for all n. Suppose now that we work with a linear

programming relaxation of Pl utilizing a subset of the facets of P, given in Theorem

1. Specifically, suppose that we have identified all cliques of G that are of maximum

cardinality (this is in general a proper subset of all cliques of anti-holes). Denote by

1

1

I

M.W. Padberg

432

A the corresponding clique-node incidence matrix. Then A = CP, R ‘ . Let P =

t?, x 2 0) be the linear programming relaxation of the node-packing

problem on G. Then PI C P C P. As one readily verifies, max {Z,”=,xi x E P } =

2 + l/[n/2] and the integer optimum of 2 follows by simply rounding down.

The interesting fact exhibited by the example is that the knowledge of merely a

few of the facets of PI in the case of odd anti-holes permits one to obtain a bound on

the integer optimum that is “sharp” as compared to the bound obtained by working

on the linear programming relaxation involving the edge-node incidence matrix of

the anti-hole (which is arbitrarily bud according to how large one chooses n ) . The

general question raised by this example is of course, how often (in a statistical

sense) it will be sufficient to work with only a small subset of all facets of a set

packing polyhedron PI (such as those given by cliques, holes, etc.) in order to verify

s-optimality of some extreme point of PI with respect to some linear form cx,

where F is some given tolerance-level measuring the distance of an 1.p. optimum

from the true integer optimum objective function value.

1

{x E R” A x S

I

Acknowledgement

I am indebted to E. Balas and L.E. Trotter, Jr. for helpful criticism of an earlier

version of this paper. In particular, Les Trotter pointed out to me an error in the

original proof of Theorem 6.

Appendix

As Theorem 4 asserts more than proven in , we shall provide a proof of the

new part in Theorem 4, which states that the complement W ( n , k ) of a facetproducing web W ( n , k ) strongly produces the facet c;=,x,s h if and only if

n = kh + 1, where h = [ n / k ] .We first prove the only-if part of the sentence. To do

so, it suffices to show that the web W ( n ,k ) contains a (properly smaller) facetproducing web W ( n ’ ,k ’ ) with [ n ’ l k ’ ]= h if k 3 2, n and k are relatively prime and

n = k h + j with 2 ~ j ~ k - 1Let. k ’ = [ k / j J + l and n ‘ = k ’ h + l . Obviously,

k ’ > 2 and g.c.d. ( n ’ , k ‘ ) = 1. To see that W ( n ’ ,k ’ ) is a (vertex-induced) subgraph of

W ( n ,k ) , we check the necessary and sufficient conditions for containment of

Theorem 4 of  which require that (i) n k ’ z n‘k and (ii) n ( k ’ - 1) n ’ ( k - 1). (i)

follows because [ k / j ]+ 1 3 k / j . (ii) follows because h ( k - k ’ ) + k - 1 - j [ k / j ] 3.0.

The latter holds because g.c.d. (n , k ) = 1 implies k - j [ k / j ] 3 1. Since W ( n ’ ,k ’ ) is

contained in W ( n ,k ) , the complement W ( n , k ) of W ( n , k ) contains a subgraph

defining t h e facet C x, s h where the summation extends over a proper subset of all

x, s h is not produced by W ( n , k ) . To

vertices of %(n, k ) . Hence the facet

prove the if-part of the above sentence, we note that the vertex-sets C, =

{ i , i + k , . . ., i + ( h - 1)k) define maximum cliques in W ( n ,k ) where i = 1,. . ., n and

c;=,

Complexity of set packing polyhedra

433

indices are taken modulo n. Let B be the incidence matrix of these cliques and note

that B A T = E - R where A is the incidence matrix of all cliques in w ( n , k ) (see

), E is a matrix of ones and R is a permutation matrix. To prove that B contains

all maximum cliques of W ( n ,k ) let b be the incidence vector to any maximum

clique of W ( n ,k ) . Then bB-’ = e T - b A T R T s0 implies that bx 1 is inessential in

defining P = {x E R” Bx s e, x 3 0) or alternatively, identical to one of the rows of

B.(The vector e is the vector of n ones.) Hence, since P contains the set-packing

polyhedron associated with W ( n ,k), B contains the incidence vectors of all

maximum cliques of W ( n , k ) .Using an argument entirely analogous to the one

used in the proof of Theorem 2 of (241, one shows that the matrix B is irreducible

xi d h if n = kh + 1.

and hence, by Theorem 3, w ( n , k ) produces the facet

I

x.i”=l

References

[ 11 E. Balas, Facets of the knapsack polytope. MSRR No.323, Carnegie-Mellon University, September

1973. Forthcoming in Math. Programming.

 E. Balas and R. Jeroslow, Canonical cuts on the hypercube, SIAM J. on Appl. Math., 23 (1972)

61-69.

 E. Balas and M.W. Padberg, Set partitioning, in: B. Roy (ed.), Combinatorial Programming:

Methods and Applications, (Reidel Publishing Company, Dordrecht, 1975).

 E. Balas and E. Zemel, All the facets of the knapsack polytope, MSSR No.374, Carnegie-Mellon

University, 1975.

[4a] E. Balas and E. Zemel, Critical cutsets of graphs and canonical facets of set packing polytopes,

MSSR No. 385, Carnegie-Mellon University, 1976.

 V. Chvital, On certain polytopes associated with graphs. CRM-238, University de Montreal,

October, 1973. Forthcoming in the J. Comb. Theory.

 J. Edmonds, Covers and packings in a family of sets, Bull. A m . Math. Soc., 68 (1962) 494-499.

 J. Edmonds, Path, trees and flowers, Canadian J. Math., 17 (1965) 449-467.

 J. Edmonds, Maximum matching and a polyhedron with 0, 1 vertices. J. Rex National Bureau of

Standards, 69B (1965) 125-130.

 D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra. Math. Programming, 1 (1971)

168-194.

[lo] R. Garfinkel and G.L. Nemhauser, A survey of integer programming emphasizing computation and

relations among models, in: T.C. Hu and S.M. Robinson, eds.: Mathematical Programming

(Academic Press, 1973).

[ l l ] F. Granot and P.L. Hammer, O n the use of boolean functions in CL1 Programming, O.R.

Mimeograph No. 70, Technion, 1970.

 P.L. Hammer, E.L. Johnson and U.N. Peled, Facets of regular 0-1 polytopes. CQRR 73-19,

University of Waterloo, October, 1973.

 F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969).

 E.L. Johnson, A class of facets of the master 0-1 knapsack polytope, Thomas J. Watson Research

Center Report RD-5106, IBM Research, October 1974.

 R.M. Karp, Reducibility among combinatorial problems, in: R.E. Miller et al., eds., Complexify of

Computer Computations (Plenum Press, New York, 1972).

 G.L. Nemhauser and L.E. Trotter, Properties of vertex packing and independence system

polyhedra. Math. Programming, 6 (1974) 48-61.

 M.W. Padberg, Essays in integer programming. Ph.D. Thesis, Carnegie-Mellon University, May,

1971.

 M.W. Padberg, On the facial structure of set packing polyhedra. Math. Programming, 5 (1973)

199-2 15.

434

M.W. Padberg

 M.W. Padberg, A note on zero-one programming. Operations Res., 23 (1975) 833-837.

 M.W. Padberg, Perfect zero-one matrices. Marh. Programming, 6 (1974) 180-196.

 M.W. Padberg, Almost integral polyhedra related to certain combinatorial optimization problems,

GBA Working Paper 75-25, 1975, New York University. Forthcoming in Linear Algebra and its

Applications.

 H.M. Salkin and J. Saha, Set covering: uses, algorithms results, Technical Memorandum No. 272,

Case Western Reserve University, March 1973.

 L.E. Trotter, Solution characteristics and algorithms for the vertex packing problem. Technical

Report No. 168, Operations Research, Cornell University, 1973.

 L.E. Trotter, A class of facet producing graphs for vertex packing polyhedra. Discrete Math., 12

(1975) 373.

 H. Weyl, Elementare Theorie der konvexen Polyeder, Comm. Math. Helv. 7, 1935, 290-306

(translated in Contributions to the Theory of Games, Vol. I, 3-18, Annals of Mathematics Studies,

No. 24, Princeton, 1950).

 L. Wolsey, Faces for linear inequalities in zero-one variables. CORE Discussion Paper No. 7338,

November, 1973.

 L. Wolsey, Oral Communication, Bonn, September 1975.

 E. Zemel, Lifting the facets of 0-1 polytopes, MSRR No. 354, Carnegie-Mellon University,

December 1974.

Annals of Discrete Mathematics 1 (1977) 435-456

@ North-Holland Publishing Company

PROPERTIES OF FACETS OF BINARY POLYTOPES

Uri N. PELED

Department of Mathematics, University of Toronto, Toronto, Ont., Canada

Properties of facets of full-dimensional polytopes P with binary vertices are studied. If Q is

obtained from P by fixing some of the binary variables, then the facets of P that reduce to a given

facet of Q are determined by the vertices of a certain polyhedron V.The case where V has a

unique vertex is characterized. If P is completely monotonic and the facet of Q has 0-1

coefficients, then the vertices of V lie in a hypercube of side I, and the integer vertices correspond

to the sequential lifts or extensions. The self facets, i.e. hyperplanes spanned by binary points, are

connected to the hyperplanes spanned by non-negative integral points. Every threshold function

can be labelled by its Chow parameter vector. The faces of the convex hull of all n-argument

parameter vectors are characterized. This leads to a necessary and sufficient condition for a

parameter vector to label a self dual threshold function having a self facet separator.

1. Introduction

This paper deals with the facets of full-dimensional polytopes with binary

vertices, i.e. the convex hulls of feasible solutions of binary programming problems.

Section 2 is a unification and generalization of previous results by several authors

on the connection between facets of such a problem P and the facets of a

subproblem Q obtained by fixing some of the variables of P to binary values. The

facets of P that reduce to a given facet of Q (“lift/extensions”) are shown to be

determined by the vertices of a certain polyhedron V, and the cases where V has

only one vertex are characterized. Section 3 makes the further assumption that P is

completely monotonic (a class that subsumes knapsack problems) and that the facet

of Q has binary coefficients. The vertices of V are then shown to lie within a

hypercube of side 1, and the integral vertices correspond precisely to the facets of P

that can be obtained by “sequential” lifts or extensions. In Section 4 we examine

the totality of facets of full-dimensional polytopes with binary vertices (“self

facets”). They are shown to be connected to hyperplanes spanned by non-negative

integral points. In Section 5 we reverse the point of view and ask what threshold

functions have self facet “separators”. Every threshold function (and some other

Boolean functions) can be labelled by its Chow parameter vector. We characterize

the non-empty faces of the convex hull of all n-argument Chow parameter vectors.

The characterization of vertices and edges leads to a necessary and sufficient

condition for a Chow parameter vector to label a self dual threshold function with a

self facet separator.

435

436

U.N. Peled

2. Lifts and extensions

For an index set N = (1,.. ., n ) , let S C B N (B denotes the set (0, I}) be a set of

0-1 N-vectors. Let N be partitioned into disjoint sets U, Z, F. Then by S," we mean

the subset T C B F defined so that x E T if and only if the point y given by

yj

=

1

jEU,

0

jEZ,

xi

j E F,

i

is in S. Thus S g is obtained from S by fixing the components indexed by U and 2

to 1 and 0, respectively, and then taking only the F components of the points of S

satisfying these conditions. When U or 2 are empty we use the short notation Sz

or S u . If S is the set of feasible solutions of some 0-1 programming problem, or in

short a problem, then S," corresponds t o the subproblem obtained by fixing xi,

j E U U 2 as above. An important class of problems is that of the monotone ones.

S is monotone if whenever x E S and some components of x are changed from 1 to

0, the resulting point is still in S. In this section we relate the facets of conv (S) and

conv ( S g ) (conv denotes convex hull).

A linear inequality is said to be valid for a set of points when it is satisfied by all

points in the set, and to support the set if in addition some points of the set satisfy it

with equality. Clearly an inequality is valid for (supports) a polytope if and only if it

is valid for (supports) the set of its vertices (a polytope is a convex hull of a finite set

of points).

Definition 1. Let S g be non-empty and let

C ajx,

ao

j€F

be a valid inequality for S,". For each subset Z' C Z, the extension coefficient ez (of

(1) relative to 2 ' )is defined by

where the maximum above is - m if n o x satisfies the condition. Similarly for each

subset U' C U, the lift coefficient lu., is defined by

=

m a

::;:s

C ajxi - ao,

where the maximum above is

Proposition 1.

(1) e z , s O ;

(3)

IEF

--co

if n o x satisfies the condition.

Let S be monotone, S,"# 0 and (1) valid for S,". Then

Properties of facets of binary polytopes

437

(2) lu. is finite;

( 3 ) if (1) supports S,", then l u t S O .

Proof. By monotonicity S,"_"z.'C S,", and so

which proves (1). Similarly S,";:: 2 S,"# 0, and so

which proves (2). Moreover, if (1) supports S,", the last right-hand side is ao,which

proves (3). 0

The extension and lift coefficients impose conditions on the coefficients of valid

inequalities for S that reduce back to (1)under the substitution xi = 1,j E U, x, = 0 ,

j E Z.

Proposition 2.

Let S," be non-empty and let (1) be valid for it. If the inequality

C a,x, s a o + C a,

,EN

(4)

j€U

is valid for S, then for each Z ' C Z, &,. aj s ezr and for each U ' C U ,

lu2, In particular a, s ej for all j E Z and a, 2 1, for all j E U.

cjEu.

aj 3

Proof. T o prove ZjEz.

a, S e,., we may assume that ez' is finite. Therefore there

exists a point x E S,"_",".'satisfying ez, + CjeFa,x,= ao.But since S,"?'Z,' is a subproba, + cjEFu,xj

s ao. The

lem of S and (4) is valid for S, x must also satisfy Cjazo

bound for lifts is proved similarly. 0

We can prove the converse of Proposition 2 for pure extensions ( U = 0)or pure

lifts (2 = 0).

Proposition 3. If the inequality CJEN-,

aJx,s a. is valid for Sz, and if & E Z , a,

ez, holds for each Z' C Z, then ZJENa,x,s a. is valid for S. Similarly if ~,EN-,,aJxJs

a. is valid for Su, and if & E U , a, 3 lu holds for each U ' c U, then ~ , E N a Jsx J

a. + Z, a, is valid for S.

1

Proof. Let x E S and let Z' = { j E Z xj = 1). By (2) we have ez,+ x j E N - Z a j xCj

a,,, and since e,. 2

a, = cjEzajx,,x satisfies ~ i E N a jsx iao.The result for lifts

has a similar proof. 0

c,,,.

The preceding discussion can be generalized to mixed lift/extensions. If S," is

non-empty and (1) is valid for it, then for each Z ' C 2, U ' C U we may define the

coefficient

438

U.N. Peled

It can then be shown that

= a. - ez., cO,,,.= a.

and only if for each 2' C 2, U ' C U,

C ~ . , ~ .ao+

S

+ lu, and that (4) is valid for S

if

2 a, - 2 a,.

IEU'

,EZ'

Let us now turn to examine conditions under which (4) is not only valid for S, but

also a facet of conv(S). We recall that a polyhedron is the solution set of a finite

number of linear inequalities. Bounded polyhedra are the same as polytopes. The

dimension of a polyhedron P is one less than the maximum number of affinely

independent points of P. A face of a polyhedron P is the solution set of the system

obtained by replacing some of the inequalities defining P by equalities. In

particular, vertices are 0-dimensional faces, edges are 1-dimensional faces and

facets are faces of dimension one less than that of P. The faces of P are the same as

the extreme subsets of P and also the sets of optimal solutions of linear programs

over P. If P is full-dimensional (i.e. its dimension equals the number of variables),

then in any system of linear inequalities defining P, the irredundant inequalities

correspond precisely (up to proportion) to the facets of P. As is customary, we call

these inequalities themselves the facets of P. T o state the next result, we use the

following definition.

Definition 2. Let (1) be a valid inequality for S,". Then its valid polyhedron is

v = { a E R~~~ every x E s satisfies (4)).

By definition, V is the polyhedron whose points are the U and 2 components of

all valid inequalities for S that reduce to (1) by the substitution x, = 1, j E U, x, = 0,

j E 2.The remark following Proposition 3 gives a defining system for V in terms of

c Z..U'.

I

Proposition 4.

Proof.

The valid polyhedron is full -dimensional and unbounded.

If M is a large enough constant and

a,

=[

-M

j € Z

M

jEU,

then a E V. Thus V is not empty. Let d, be the j unit vector. We show that if

a E V, then a - d, E V for j E 2 and a + d, E V for j E U. To prove t h e first of

these statements please note that for all binary x, if x, = 0, then (4) has the same

form for a - d, as for a, and if x, = 1, then (4) for a - d, is the sum of (4)for a and

the valid inequality - x, s 0. The second statement is proved similarly.

The next theorem belongs to the type of polarity results that are obtained by

Araoz  and also by Edmonds and Griffin [private communication].

Properties of faceis of binary polytopes

439

Theorem 1. Let conv ( S ) and conv ( S g ) be full-dimensional, and let (1) be a facet

of the latter. Then (4) is a facet of the former i f and only if a = (ai,j E U U Z ) is a

vertex of the valid polyhedron V of (1). In that case (4) is called a liftlextension

of (1).

Proof. By definition, the validity of (4) means the same thing as a E V. To show

the "only if" part of the theorem, it is sufficient to prove that a is an extreme point

of V. Suppose that a = 5(b + c ) , where b, c E V . Then the two inequalities

are valid for conv(S) and (4) is their arithmetic mean. Since conv(S) is fulldimensional and (4) is one of its facets, it must coincide with (5) and (6), otherwise it

is redundant. Thus a = b = c, proving that a is extreme in V.

We now show the "if" part. This time we show that there are n = IN1 affinely

independent points of S satisfying (4) with equality, proving that it is an

( n - 1)-dimensional face. For ease of writing, let us reindex the variables so that

U = (1,. . ., r}, 2 = { r + 1,.. ., r + s } , F = { r + s + 1,.. ., n } . Since a is a basic solution of the system of inequalities (4) for all x E S, there exist r + s points

x l , . . ., x r f SE S such that a satisfies the corresponding inequalities (4) as equalities

and the coefficient matrix of a l , . . ., a,,, in these inequalities, namely

is non-singular. Also since (1) is a facet of the full-dimensional conv(S,"), there

exist n - r - s affinely independent points y'+'+', . . ., y " E S," that satisfy (1) with

equality. Let these points form the rows of the matrix

By definition of S y, the points x r C s + ' ,. . ., X " defined by

x; =

1

j€U,

0

j€Z,

Y;

iEF,

i

belong to S. They too satisfy (4) with equality. It remains to show that the rows of

the matrix

440

U.N. Peled

--I-

x;;;. ..

X'+S

x;;;+,

,+s+ 1

yr+s+1

. . . x,

. . . y ,.+ \$ + I

are affinely independent. If r + s = n, then the rows of X * differ from the rows of X

by a fixed translation (1,. . ., 1,0,. . .,O). As the rows of X are affinely independent,

so are the rows of X * . If r + s < n, subtract the last row of X * from each other

row. It is enough to show that the first n - 1 rows are now linearly independent.

These rows now constitute a matrix of the form [g :I, where the rows of X are

linearly independent and the rows of L, being the differences y""" - y", . . .,

n-' are also linearly independent. This completes the proof of

Theorem 1. 0

")

Under the conditions of Theorem 1, suppose further that 2 contains an index i

such that the extension coefficient e, relative to (1) is finite. If we consider S y as a

subproblem of S,"-,, then the valid polyhedron V is I-dimensional with a vertex at

e,. Thus the inequality &EFa,x,+ e,x, S a,, is a facet of conv ( S g - , ) . If Z - i contains

a further index whose extension coefficient relative to the present inequality is

finite, the process can be continued. This is called sequential extension of (1). In

particular, if S is full-dimensional and monotone, so are all its subproblems of the

form Sz, and by Proposition 1 each facet of such a subproblem can be sequentially

extended to (one or more, depending on the order of extension) facets of the

complete problem. Hence, by Theorem 1, V has in fact vertices in that case. In a

similar way one also has sequential lifts and sequential liftlextensions. Sequential

extensions have been studied by many authors, including Balas , Balas and

Zemel , Hammer, Johnson and Peled , Nemhauser and Trotter , Padberg

, Pollatschek , Trotter , Wolsey  and Zemel . Sequential lifts are

treated by Wolsey , in a work that stimulated my interest in lifts. Theorem 1 was

proved by Zemel  for the case of pure extensions. Non-sequential extensions

are also discussed by Balas and Zemel .

We conclude this section with two corollaries and an example of Theorem 1.

I

Corollary 1. Under the conditions of Theorem 1, i f U U Z 1 s 2, then every vertex

of the valid polyhedron V corresponds to a sequential liftlextension of (1).

I

I

Proof. We have already considered the case 1 U U 2 = 1. For 1 U U Z = 2,

consider the typical case of pure extensions, U = 0, 2 = {I, 2}, other cases being

Properties of facers of binary polytopes

44 1

similar. We then have V = {(a,, a,)/ a , s e l ,az S e z , a l + azs e l z } . It is easy to

verify that if e l + ez S e l > ,then V has a unique vertex ( e l ,ez), and if e l + ez > e12,

then V has two vertices ( e l ,e l z- e l ) and ( e l ,- e z , e z ) . All these vertices represent

sequential extensions: in the first case the two sequences commute (give the same

facet) and in the second case they do not. 0

Corollary 1 appears, for pure extensions, in Hammer, Johnson and Peled  and

in Zemel .

Corollary 2. Under the conditions of Theorem 1, assume further that all e,, j E Z

and l,, j E U are finite. Then the following two conditions are equivalent:

(1) the inequality

is valid for S ;

(2) the valid polyhedron V has a unique vertex (i.e. (1) has a unique

liftlextension ).

In that case the unique vertex is in fact (e,,j E Z ; l,, j E U ) .

Proof. Please note that as e, and 1, are finite, V has vertices (it being contained in

an orthant of RU"").

(1) =3 (2). It is enough to show that whenever (4) is a facet of conv(S), a, = e,

for j E 2 and a, = 1, for j E U (this follows from Theorem 1). Since (4) is valid for

S, Proposition 2 gives a, < e, for j E Z and a, 3 1, for j E U. Therefore we can add

the valid inequalities

(a, - e)x, < 0

(a, - l,)x,

i E Z,

s a, - 1, j E U

to the valid inequality (7) to obtain (4). But (4) is a facet of the full-dimensional

conv ( S ) , and so it is irredundant. It follows therefore that a, = e, for j E Z and

a, = I, for j E U.

(2)

(1). Note that by Theorem 1 there is a unique facet of conv(S) of the

form (4). On the other hand, such facets can be obtained by sequential

lift/extensions. The sequence may start from any j E U U Z, since the lift/extension

coefficients are all finite. This yields a, = 1, if j E U and a, = e, if j E Z. Therefore

(7) is the unique facet in question and (1) certainly holds. 0

=+

Special cases of Corollary 2, involving pure extensions, appear in Balas ,

Hammer, Johnson and Peled  and Balas and Zemel .

Example. Let S be the set of incidence vectors of the node packings of the

~ 1 (indices modulo 5). S is

pentagon, i.e. the vectors x E B 5 such that x, + x , + s ### Tài liệu bạn tìm kiếm đã sẵn sàng tải về

Chapter 20. Complexity of Machine Scheduling Problems

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