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Chapter 33. On Antiblocking sets and Polyhedra

# Chapter 33. On Antiblocking sets and Polyhedra

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J. Tind

508

2. Antiblocking sets

Let B C R" be a closed, convex set, containing 0. The polar set B * of B is defined

as

I

B* = { X * E R " x

.X*S

i,vx E B}.

B * is also a closed, convex set that contains 0.

Additionally we have that B ** = B, i.e. B is again the polar set of B *. This is the

Minkowski polarity correspondence (see e.g. [5, section 141).

Let also D C R" be a closed convex set containing 0. Define the antiblocking set

B C R" of B with respect to D as follows:

B=B*no.

In the following we will investigate conditions under which

B = B,

(2.1)

i.e., when B is the antiblocking set of B with respect to D. In that case B and B are

called a pair of antiblocking sets.

It is seen that with D = R" we are back in the Minkowski polarity. But in more

general cases it is necessary to impose special conditions on B in order to show the

polarity correspondence in (2.1).

Let c l C denote the closure of C and let c o n v C denote the convex hull of C,

where C C R". We then have the following theorem which gives a necessary and

sufficient condition for equation (2.1) to be valid.

Theorem 2.1.

B = B, if and only if B = cl(conv(B U D * ) )f

D.

l

Proof. B = (B)*n D = (B*n D)* n D.

We have for polar sets in general that

(B n D ) * = c l c o n v ( B * U D * )

(see [ 5 , Corollary 165.21). Hence with B replaced by B * we get that

E = cl(conv(B** u D * ) )n D

= cl (conv(B u D *)) n D.

The next theorem gives another set of conditions that are necessary and sufficient

for (2.1) to be valid. These conditions can especially be applied when B is described

as the intersection of halfspaces.

Theorem 2.2.'

= B, if and only i f there exists a closed, convex set C C R"

containing 0 such that B = C n D and such that C* C D.

For polyhedra, Theorem 2.2 is a special case of joint work by Jnlian Arloz, Jack Edmonds and

Victor Griffin. Personal communication.

On antiblocking sets and polyhedra

509

Proof. Let us first assume that = B,and let C = cl (conv (B U D *)). Obviously,

C is closed, convex and 0 E C. Theorem 2.1 implies that B = C n D. Additionally,

as C 2 D *, we have that C* D ** = D. This shows one direction of the theorem.

Now assume that we have a set C such that B = C f l D and C * C D. [It is

remarked that C here in general might be different from the previous set

cl(conv(B U D * ) ) ] .From theorem 2.1 it is now sufficient t o show that

B

= cl (conv (B

u D *)) n D.

Since B = B fl D C cl(conv(B U D * ) )fl D it is enough to show the reverse

inclusion:

B 2 cl (conv (B u D *)) n D.

(2.2)

c

By assumption C* D, which implies that C 2 D * . Moreover, C 2 B. Since C is

closed and convex, we obtain that

C 2 cl (conv (B U D *)). Hence

cl(conv(B U D * ) )fl D C n D = B,where the last equation follows by assumption. This shows (2.2), and the theorem is proved.

c

The assumption C* C D in the theorem expresses in particular that all supporting hyperplanes for C have their normals contained in D. (The defining linear

forms are normalised ( = 1)).

If D = R : = { x E R " I x a O ) and C = { X E R " ) A X S ~ }where

,

A is an m x n

matrix of nonnegative elements and 1 = (1,. . ., 1) with m elements, then the

theorem can be applied on B = C n D. In this case B and B constitute a pair of

antiblocking polyhedra [l].

Theorem 2.2 is an extension of a result in [6].

A similar discussion can also b e made for blocking sets and polyhedra ([6]

and [7]).

3. A geometrical illustration of antiblocking sets

The relation between B and its polar set B * can be given in the following

equivalent way.

1

B * = { x * E R" (x, l ) - ( x * ,- 1)S 0, VX E B},

where (x, 1) and ( x * , - 1) are vectors in R"+'.

Hence, if we consider the space R"+' and let B be placed in the hyperplane If+',

where H" = {(x, 1) x E R"}, then B * can be obtained as follows. Construct the

cone P generated by B with vertex at O E R"" and its polar cone P * =

{y * E R"" y . y * S 0, Vy E P } . Where B * intersects the hyperplane H-' =

{(x, - 1) E R"" x E R"} we get an image of B *. B is now by definition obtained as

the intersection of B* and D.

Let us look at the situation where D = R: = {x E R2 x 3 0). Fig. 1 gives now an

I

I

1

1

510

J. Tind

H

+'

H -1

B

and

B*and

are indicated by

(B)"

FRfl

are i n d i c a t e d by

Fig. 1.

=

ll

illustration of a set B E R: and its antiblocking set

= B. Here B and B are actually polyhedra.

B E RZiI1 me situation

where

4. An economic interpretation

Consider a concave, nonnegative closed function f ( x ) : R? +R,.

Let sub, f E R" denote the nonnegative subgraph of f ( x ) , i.e.

I

sub+f = {(x, y ) E Rnfl x

3 0 , 0 =zy S f ( x ) } .

Since f(x) is a nonnegative function, it is uniquely determined by sub+f.

With the given specifications on f ( x ) it follows that sub, f is a closed, convex set,

containing 0.

Define the following function f ( x *) :R: +R,.

511

O n antiblocking sets and polyhedra

f ( x * ) = sup{y * E R

1 -x

*

x * + y * f ( x ) 5z 1, v x 5 0).

Call f ( x *) the antiblocking function of f . f ( x *) becomes also nonnegative, and its

nonnegative subgraph is given by

sub+f = {(x *, y *) E R""

I

(X

*, y *) . ( -

X,

y)

1

1

for all (x, y ) E sub+f}fl {(x *, y *) E R"" (x *, Y *) 3 0).

This shows that f is concave and closed.

Let T denote the linear transformation T : (x, y ) + ( - x, y). If D = R?+',it is

seen that sub+f is obtained by the antiblocking relation with respect t o D as

follows:

sub+f = T(sub+f).

Additionally it is assumed that f ( x ) is a non-decreasing function in each

component, which implies that all supporting hyperplanes for T(sub+ f ) have their

normals in D. Hence, it is obtained by the same reasoning as in the proof of

theorem 2.1, that

T(sub+ f ) = sub+f.

This shows that

7 = f,

(4.1)

i.e., f is the antiblocking function of f .

We will try to give an economic interpretation of this equataion in the following.

Note that f ( x * ) can be expressed alternatively as

-

x*.x+1

f ( x * ) = inf -~

xso

f(x) '

where (x * . x + l)/f(x) = 00, if f ( x ) = 0.

The polarity will not be disturbed by rescaling, i.e. by replacement of the number

1 by an arbitrary number k > O , which means that

Assume now that a manufacturer produces a product by means of n activities.

Let the components of the vector x S O denote the activity level of each activity.

With a given activity level he produces f ( x ) units of the product. Assume

additionally that the components of the vector x * > 0 denote market prices that

equal the cost for use or consumption of one unit of the corresponding activities.

Hence x . x * is a cost for production of f ( x ) units of the product. In addition to this

cost, which is linear in x, there is supposed to b e a constant cost of size k. It is

further supposed that the manufacturer's objective is to minimize the average cost

per unit produced, i.e., the manufacturer wants to solve the following problem

512

J. Tind

Hence the antiblocking function of f denotes the minimal average cost, given a

price vector x * .

Now the manufacturer also considers selling his activities at a given level x 5 0 to

a contractor, who in return should pay him with an amount of the finished product.

For that purpose the contractor quotes a unit price x * 2 0 on each activity. Based

on this price the manufacturer at least would demand an amount of the finished

product that equals the estimated production costs, divided by the average cost per

unit, i.e.

Hence, the contractor, seeing no reason t o return more than that amount, will get

the task to find a price x * 2 0 that solves the problem

x .x*+ k

T ( x ) = inf -x.20

f(x*)

By (4.1) we get the reasonable result that with such a price the amount of finished

product does not depend on whether he produces by himself o r lets the contractor

d o it for him.

It is remarked that the idea of antiblocking functions is almost the same as the

idea of polar functions in [5, section 151. But again, the polarity is considered with

respect to a given set, here R:. This has the effect that the prices x * are

nonnegative, and the function f ( x ) is non-decreasing, which seems reasonable in

the economic context above.

5. Bounds for set-covering problems

From the preceding discussion it is seen that the antiblocking relation itself is

developed over the continuous space R". But historically the concept came u p

through studies of discrete problems, especially certain integer programming

problems [l].

In the following discussion one of these integer programming problems will be

examined, in view of the duality for antiblocking polyhedra. An example will be

given, which illustrates how one may obtain computationally simple bounds for the

value of those problems. The idea for construction of these bounds has previously

been developed and used in [4]for the set-partitioning problem, and the following

material is highly related to this work.

Here we will look at the following set covering problem:

On antiblocking sets and polyhedra

513

min 1 . x

Ax 3 w

x

30

(5.1)

and integer,

where A is an m X n matrix of zeros and ones, w is a nonnegative integer

m-vector, and 1 = (1, .. ., 1) with n elements. By removal of the integrality

requirement we get:

min1.x

Ax

2

(5.2)

w

x 30,

which, as usual, by standard L P gives a lower bound for the objective function in

(5.1). But in some cases a bound can be obtained even simpler. For example, if the

columns in A are incidence vectors for all maximal chains in an oriented network

without cycles, then the problem can be solved by an algorithm of the network flow

type. For instance: Connect all endpoints of the chains t o a source and a sink,

respectively. Place a lower bound of w , (the ith component of w ) on the ith node,

and compute the minimal flow from s to t.

With w = (1,. . ., 1) this problem is a generalization of o n e part of the Dilworth

theorem. See for example [3]. T h e result is integer. This is seen directly, or in more

general terms from the antiblocking theory this follows by the min-max equality,

are the incidence vectors of all

which here holds for A and A. The columns in

maximal antichains in the same network, and the set B = { y 3 0 yA l},(which is

the dual constraint set of (5.2)) and the set B = {y * 3 0 y *A S 1) constitute a pair

of antiblocking polyhedra. See [l].

Now generally A is not the incidence matrix of all maximal chains in a network.

But a network can be constructed in which A is the incidence column matrix of at

least some chains. Then, by solving the covering problem over all chains, we receive

a lower bound for (5.1). This lower bound is easy to compute, although in general it

is weaker than the bound obtained by solving (5.2).

Consider the following example, where w = (1,.. ., l), [4]:

I

4

m i n z xi

i=l

I

J. Tind

514

Let the nodes in the network be numbered corresponding t o the row numbers of

A. Then the network looks as follows,

5

and the bound is 2 (the minimal number of chains that cover all nodes, which is

equal t o the maximal size of an antichain; Dilworth).

The result is generally dependent on the permutation of rows. For example, with

the matrix:

we get the network

1

3

2

o

:

c

:

=

4

:

=

5

0

and the bound is equal t o 1.

We can also construct a loopless oriented network, in which the matrix A

corresponds t o some of the antichains in the network. For the problem (5.3) the

network may look like the following:

l

5

The minimal number of covering antichains, which is equal to the largest chain

(the companion t o the Dilworth theorem [l])gives a lower bound. T h e result here

is 2.

An upper bound for the set covering problem can be found in a similar way by

network flow methods, where now the rows of the matrix are incidence vectors for

chains (or antichains). For example, with chains we get the following network for

the problem (5.3):

t

;

,

n

,+-.',

\

I

/

I',

J'

S

/

'a

/

9

On anriblocking sets and polyhedra

515

The numbers correspond to the columns. The endpoints of the chains are

connected to a source s and a sink t, respectively. The problem is now to find a

minimal number of nodes that block all s-t chains, (which is equal to the maximal

number of node independent chains from s to t ; Menger’s Theorem). Here the

result is 2.

We believe that such bounds may be helpful in an algorithm for solution of set

covering type problems, and an algorithm incorporating that feature is now under

development.

Acknowledgement

I wish to thank J. Ar6oz for his useful comments on an earlier version of this

paper.

References

[l] D.R. Fulkerson, Anti-blocking polyhedra, J. Comb. Theory 12 (1972) 50-71.

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

168-194.

[3] D.R. Fulkerson, Flow networks and combinatorial operations research, A m . Math. Monthly 73

(1966) 115-138.

[4] G.L. Nemhauser, L.E. Trotter, Jr. and R.M. Nauss, Set partitioning and chain decomposition,

Management Sci. 20 (1974) 1413-1423.

[5] R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1970).

[6] J. Tind, Blocking and antiblocking sets, Math. Programming 6 (1974) 157-166.

[7] J. Tind, Dual correspondences for blocking and antiblocking sets (1974), 10 pp., Institut for

Operationsanalyse, University of Aarhus; or Report No. 7421-OR, Institut fur Okonometrie und

Operations Research, University of Bonn.

[S] A.C. Williams, Nonlinear activity analysis, Management Sci. 17 (1970) 127-139.

Annals of Discrete Mathematics 1 (1977) 517-525

@ North-Holland Publishing Company

ON THE GENERALITY OF MULTI-TERMINAL

FLOW THEORY

L.E. TROTTER, Jr.*

Department of Operations Research, College of Engineering, Cornell University,

Ithaca, NY, U.S.A.

We consider the problem of determining maximal flows between each pair of nodes in an

undirected network. Gomory and Hu have studied this problem and have provided an efficient

algorithm for its solution. W e reexamine their procedure and generalize certain results of

multi-terminal flow theory using well-known aspects of matroid theory. Additional implications

afforded by this approach are also discussed.

1. Introduction

In their interesting paper [5] (see also [4,6]) Gomory and H u have considered the

problem of determining the maximum flow value between each pair of nodes in a

finite, undirected graph. This problem, known as the multiterminal maximum pow

problem, has also been studied by Mayeda [9] and Chien [l].In [3] Elmaghraby has

examined the sensitivity of multi-terminal flows t o changes in the capacity of a

single edge in the graph. In the present paper we adopt the viewpoint of matroid

theory and reexamine some basic results of multi-terminal flow theory in this more

general, abstract setting. W e begin with a brief summary of multi-terminal flow

theory. In this discussion reader familiarity with the fundamental aspects of

network flow theory, as set forth in [4], is presumed.

Assume given a finite, undirected graph (network) G. W e will further require

that G has neither loops nor multiple edges and that G is connected, though these

latter assumptions are only for convenience of exposition. As usual, associated with

each edge e of G is a nonnegative, real-valued capacity c ( e ) . W e also have, for

each unordered pair of nodes {x, y } of G, a maximum flow value v ( { x , y})' between

x and y with respect to the given edge capacities. The real-valued, nonnegative

function u is called the flow function for G. Notice that when G has n nodes u may

be viewed as a function defined on the edges of K,, the complete graph on n nodes.

Our primary concern is with the flow function 21. O n e question of interest is that

* This research was partially supported by grant GK-42095 from the National Science Foundation to

Yale University.

The cumbersome notation is chosen t o emphasize th e fact that u is a function from the pairs of

nodes of G to the nonnegative reals. Th e reason for this emphasis will become apparent in Section 3.

517

L.E. Trotter, Jr.

518

of realizability : When is a function the flow function of some graph? Gomory and

Hu [5] have answered this question with the following characterization.

Theorem 1. A function u from the edges of K , to the nonnegative reals is the flow

function of an n-node undirected network if and only i f

u({xl, x,})

3 min

for any node sequence

[u({x1,

x2,. . ., x,.

XI,

u((x2, x3),

~ 2 1 ) ~

. . .) ~ ( { X , - I ~xp})19

(1)

0

Two networks which have the same flow function are termed flow-equivalent. An

important consequence of (1) which becomes evident in the construction used to

prove the sufficiency of these conditions is that every undirected network is

flow-equivalent to a tree. Thus the flow function for a graph with n nodes assumes

at most n - 1 different values.

A second question of interest is the following: How does one efficiently

determine the flow function for a given graph? Of course, one may construct the

flow function for an n-node network by solving each of the (2n) maximum flow

problems which correspond to all pairs of nodes in the network. However, since the

flow function assumes at most n - 1 distinct values, one might hope to d o better.

Gomory and Hu [5] have accomplished this by providing an elegant algorithm

which determines the flow function by solving only n - 1maximum flow problems.

In order to describe their procedure we use the max-flow min-cut theorem of

Ford and Fulkerson [4] to change emphasis slightly and view v ( { x , y } ) as the

capacity of a minimum cut separating x and y . If sets X , partition the nodes of G,

we denote the corresponding cut by

x

x)= { e = {x, f }: x E X , 3 E x

(X,

and e is an edge of G}.

x

When each of the sets X n Y , X n ?, fl Y, n is nonempty, the two cuts

( X ,2)and ( Y , F) cross each other; otherwise these cuts are non-crossing.A family

of cuts is termed non-crossing if each pair in the family is non-crossing. The

following result which appears in [7] characterizes families of non-crossing cuts.

Lemma 1. In a graph on n nodes, the families of n - 1 non-crossing cuts correspond

precisely to the spanning trees of K..

Certain of the minimum capacity cuts in a network also obey a non-crossing

property. This is demonstrated in the following lemma, which is a simple

consequence of the results of [5].

Lemma 2. Suppose cuts (x1,

g1),

. . ., ( & I, % I ) are non-crossing and (x,X i ) is

a minimum capacity cut separating xi and I,for 1 < i < k - 1. Also assume that no

(xi,Xi)separates x k and T ~ .Then there exists a minimum capacity cut ( X k j x k )

separating X k and ,fk which crosses no (xi,

1 C i C k - 1. 17

x),

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