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 nondecreasing 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 nondecreasing, which seems reasonable in
the economic context above.
5. Bounds for setcovering 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 setpartitioning 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
mvector, 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 minmax 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 st 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, Antiblocking polyhedra, J. Comb. Theory 12 (1972) 5071.
[2] D.R. Fulkerson, Blocking and antiblocking pairs of polyhedra, Math. Programming 1 (1971)
168194.
[3] D.R. Fulkerson, Flow networks and combinatorial operations research, A m . Math. Monthly 73
(1966) 115138.
[4] G.L. Nemhauser, L.E. Trotter, Jr. and R.M. Nauss, Set partitioning and chain decomposition,
Management Sci. 20 (1974) 14131423.
[5] R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
[6] J. Tind, Blocking and antiblocking sets, Math. Programming 6 (1974) 157166.
[7] J. Tind, Dual correspondences for blocking and antiblocking sets (1974), 10 pp., Institut for
Operationsanalyse, University of Aarhus; or Report No. 7421OR, Institut fur Okonometrie und
Operations Research, University of Bonn.
[S] A.C. Williams, Nonlinear activity analysis, Management Sci. 17 (1970) 127139.
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Annals of Discrete Mathematics 1 (1977) 517525
@ NorthHolland Publishing Company
ON THE GENERALITY OF MULTITERMINAL
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
multiterminal flow theory using wellknown 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 multiterminal 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 multiterminal flow theory in this more
general, abstract setting. W e begin with a brief summary of multiterminal 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, realvalued 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 realvalued, 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 GK42095 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 nnode 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 flowequivalent. 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
flowequivalent 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 nnode 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 maxflow mincut 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 noncrossing.A family
of cuts is termed noncrossing if each pair in the family is noncrossing. The
following result which appears in [7] characterizes families of noncrossing cuts.
Lemma 1. In a graph on n nodes, the families of n  1 noncrossing cuts correspond
precisely to the spanning trees of K..
Certain of the minimum capacity cuts in a network also obey a noncrossing
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 noncrossing 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),