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SOME ABSTRACT PIVOT ALGORITHMS
by
Curtis Green and Thomas L. Magnanti
OR 037-74
August 1974
Supported in part by the U.S. Army Research Office (Durham)
under Contract No. DAHCO4-73-0032
Some Abstract Pivot Algorithms
Curtis Greene
Department of Mathematics
M.I.T.
Thomas L. Magnanti
Sloan School of Management
M.I.T.
Abstract:
Several problems in the theory of combinatorial
geometries (or matroids) are solved by means of algorithms
which involve the notion of "abstract pivots".
The main
example is the Edmonds-Fulkerson partition theorem, which is
applied to prove a number of generalized exchange properties
for bases.
1.
Introduction
The theory of combinatorial geometries
were first called
[18])
(or matroids, as they
concerns properties of a matrix which
depend only on a knowledge of which sets of columns are indepenThis paper concerns a number of problems and related
dent.
algorithms in combinatorial geometry which derive
analog of "pivoting" in matrices.
from the abstract
In matrix theory, a pivot is a
single application of the Gauss-Jordan elimination process, which
In abstract
eliminates one variable from a set of equations.
combinatorial geometries, the existence of pivots is assumed as
an axiom, in the form of a replacement property for bases:
S
if
x
there exists an element
S,
(bases) and
are maximal independent sets
T
and
y
(S-x) v
such that
T
y
is
a basis.
If we think of
with
S
S
and
T
as sets of columns in a matrix
an identity submatrix,
responds to a "pivot
then replacing
about position
x
x
in column
by
y
M,
cor-
y."
This replacement property allows one to recover some of
the algebraic structure of matrices in combinatorial form.
one example of this, Rota has observed
[15]
As
that many determinant
identities have "analogs" which are valid in any combinatorial
geometry.
Such results are obtained by ignoring the values of
determinants and considering only whether or not they are
zero
(i.e. whether the underlying sets of vectors constitute a bases).
For example, one can use determinants to prove the following
-2-
exchange property* for bases
(which is stronger than the replacement
axiom but follows from it):
if
y
T
S
and
T
are bases, and
such that both
(S-x)
y
x
S, there exists an element
and
(T-y)
The argument for matrices is as follows:
by an identity matrix, and
T
x
are bases.
If
S
is represented
is an arbitrary nonsingular square
matrix, then
det T =
I
+ det((S-x)~ y) . det((T-y)
x)
yeT
as can be seen by expanding det T by cofactors along "row x".
det T
Since
O0, some term on the right must be nonzero and the result
follows.
It is not hard to give a "determinant-free" proof of the exchange property
(see [21,13])
and this proof shows that the property
holds in any combinatorial geometry.
In [8]
one of the authors obtained a "multiple exchange pro-
perty" for bases, which corresponds to the Laplace expansion theorem
for determinants in the same way that the ordinary exchange property
corresponds to expansion by cofactors:
X
C
S, then there exists a subset Y
if S and T are bases, and
C
T such that (S-X)
Y and
(T-Y) J X) are both bases.
The proof by determinants is virtually identical to the one just
described when X is a singleton.
However a proof valid in any geo-
The distinction drawn here between "replacement" and "exchange" does
not correspond to standard terminology.
-3-
metry is much more difficult.
In this paper we give a new construc-
tive proof, by describing an elementary pivot algorithm for carrying
out the exchange.
We will also show how a number of results related to the multiple exchange property can be expressed as "abstract pivot theorems",
and describe the pivot algorithms associated with them.
Among other
things, we will show how Greene's exchange theorem follows immediately from the powerful "matroid partition theorem" of Edmonds and Fulkerson [7].
We describe this theorem in section 2, including an al-
gorithm which, although not essentially new, takes on a particularly
simple form in the present context.
In section 3, we describe a num-
ber of "multiple exchange theorems", all of which can be reduced to
the Edmonds-Fulkerson theorem, and hence can be proved by elementary
pivot techniques.
In section 4, we raise a new question:
can a
multiple exchange of k vectors be carried out by a sequence of k
single exchanges?
We conjecture that some permutation of the vec-
tors can be exchanged sequentially, and.prove that this is the case
for k=2.
2.
Pivot Operations and the Edmonds-Fulkerson Theorem
Recall that a combinatorial geometry G(X) consists of a finite
set X together with a collection of subsets of X called bases, such
that
and x
(i) all bases have the same size and
S, then there exists an element y
is a basis.
some basis.
(ii) if S and T are bases,
T such that
(S-x)v
A set A is called independent if it is contained in
y
-4-
If it is possible to associate the elements of X with columns of
a matrix M in such a way that bases correspond to maximal independent
sets of columns, we say that G(X) is coordinatized by M.
Examples
show that not every geometry can be coordinatized by a matrix; nevertheless
most arguments involving the elementary tools of linear al-
gebra - independence, dependence, linear closure, dimension, etc. carry over to combinatorial geometries with no difficulty.
The read-
er can safely assume that any such argument appearing in this paper
can be derived solely from the axioms for bases.
We mention two important properties:
first the rank of a subset
A, denoted r(A), is defined as the maximum size of an independent
subset of A and obeys the submodular law:
r(A V B) + r(A rf
Second, if S is a basis, and y
C(y,S) of elements x
B) < r(A) + r(B).
S, we say that y depends on the set
S such that (S-x)u
y is a basis.
More gener-
ally, we say that y depends on a set A if there exists a basis S such
that C(y,8)
C
A.
The set y C(y,S) is called the circuit deter-
mined by y and S, and is a minimal (in the sense of set-inclusion)
dependent set. Most important for our purposes is the fact that "dependence" is transitive:
if y depends on A, and every element of A
depends on B, then y depends on
.
We will make free use of these
ideas without attempting to justify our reasoning - the reader can
refer to [14] or [18]
for a detailed development.
Suppose that G(X) is coordinatized by matrix M, and S is a basis
whose columns in M are coordinate vectors.
(This means that M is in
reduced echelon form with respect to the columns corresponding to S.)
-5-
For any y
S, the elements of C(y,S) can be identified immediately
by looking at the nonzero entries in column y.
x
Each element
C(y,S) can be replaced by y to form a new basis T = (S-x)
y.
We call the operation of transforming S into T a pivot about x in y
(with respect to S).
whenever x
Whenever such a pivot is possible, that is,
C (y,S), we write
y
X.
---4
S
These symbols define a directed graph with vertex set X and a
multi-labelled set of directed edges, with one label type for each
basis S.'
In concrete terms, each pivot represents a single applica-
tion of the Gauss-Jordan elimination process (applied to the column y).
Much of this paper concerns the interpretation of these symbols in
special situations.
It will be convenient to know when a chain of pivots
x
e y- --
S
z.. ---
T
can be carried out simultaneously.
w
U
That is, if a basis appears
several times in the chain, we need conditions which guarantee that
all of the replacements involving it can be made at once.
The following lemma provides a very useful condition of this
type, which applied even when the bases
S, T,
... , U
come
fvom different geometries.
tA related structure, called a basis graph has been studied by several
authors (Bondy [1], Holzmann and Harary
O]J, Maurer [12],[13]).
The
structures are formally distinct, however, since the vertices in a
basis graph are bases, with edges defined by pivots. Here, the vertices are elements of X and each basis determines a class of edges.
Lemma 2.1
Suppose that
Yk
Y -'
respectively.
(Neither the
to be distinct.)
YO--0
B1
and
are required
Gi's
nor the
Bi's
X
G(X),...,G(X)
G 1 (X),
are bases of geometries
B, ,..,B
B
are elements of
Suppose that
Jk-l
Y---- '''
B2
Yk
Bk
Assume further that this chain is
is a chain of pivots.
minimal, in the sense that no shorter path from
exists using the labels
B1, B2LJ
,
Bk.
to
y
yk
Then each of the
sets
= (Bi-Ya-yb- ...
B
(where
Bi
= B
Yc )
=Ba ...
=
- '
Yb-1
Bc)
'
is a basis in
% Y-1
Gi(X),
i = 1,2,...,k.
B i, the pivots on elements
Proof:
We observe that, for each
of
can be carried out sequentially, provided that the last ones
Bi
are made first.
If
Bi
appears only once in the list, then
is trivially a basis (by definition of
appears more than once, then
Yi-1-
Yi )
y
can still be
B
Yi
performed unless some member of the circuit
say
yj, has been removed from
then there exists an arc
Bi
Yil1 ---
in an
Yj
C(yi
1
violates the assumption of minimal length.
Bi
Bi
earlier pivot.
with
If
B!
But
j > i, which
-7-
Next we describe the matroid partition theorem of Edmonds
and Fulkerson.
The question is this:
G 1 (X), G 2 (X),...,Gk(X)
suppose that
are geometries defined on the same set
Under what conditions is it possible to partition
B.i
such that, for each
i,
Bi
X
is independent in
.
into blocks
Gi?
Moreover,
how can one find such a partition if it exists?
In terms of matrices, the problem can be described as
follows:
having
suppose that
IXI
M 1, M2,..., Mk
columns, which are stacked on top of each other to
form a large matrix
M*.
Under what conditions is it possible
to partition the columns of
each
i, the submatrix of
columns.
Bi
determined by
so that for
Bi
has independent
G i , exists if and only if for each
r(A) + r 2 (A) +
rank of
Mi
into sets
(Edmonds, Fulkerson) A partition of X into sets
Bi, independent in
<
M*
The answer is contained in the following:
Theorem 2.2
IAI
are matrices, each
A
in
... + rk(A), where
ri (A)
A C X,
denotes the
G..
Necessity of this condition is
trivial, so it suffices
to prove that a partition exists whenever the conditions are
satisfied.
We now give an algorithm, based on pivot operations,
which shows this:
-8B1, B2 ...
Suppose that
property that
Bk
is a basis of
Bi
Gi, for each
with the
X
are subsets of
If
i.
U
Bi
=
X,
we are done, since we can form a partition into independent
If
sets by removing duplicated elements.
y
X -
U Bi
one of them.
X
We must show how to rearrange the elements of
y
B! with the same property, and add
into new sets
Bi
X, let
U Bi #
to
If this is always possible, we can continue until
is exhausted, and a partition is obtained.
The algorithm is based on a labelling procedure:
Step
(0) Label the element
y.
Step (1) For each labelled element
z
element
Step (2)
z
y' .
such that
y', label every unlabelled
for some
B..
If an element common to two bases, say
Bj, has been labelled, stop.
B.
and
1
Otherwise go back to step 1.
When the labelling procedure stops, there is a chain
) Y1 l
Y = Y
where
(2
y
y2
is common to two bases, say
yj
OF.
(j
B )
and
Bk
(It
is understood that bases can appear several times in the list.)
i = 1,...,k,
Now define, for each
Bi
3i
if
Bi
does not appear in the list
=
(Bi Ya-Yb
if
-Yc)
Ya-l U Yb-l U
B. = B(a) = B(b) =
= B (C)
'
Yc-1
-9-
From the nature of the labelling algorithm, it is clear that
the chain from
y
to
yi
is minimal.
Hence the previous
lemma applies, and it follows that each
G i.
Clearly
B
= y u
Bi
is a basis in
B i , and we have added
y
as
desired.
It remains to show that the labelling process terminates
---
that is, some element common to two bases is
eventually labelled.
Suppose to the contrary, that the
algorithm proceeds until Step (1) no longer labels anything new.
If we denote the set of labelled elements by
L n
depends on
L
Bi
in each geometry
are disjoint.
57
since
Bi
y
r
L
(L)
Bil
y
L
G i, and the sets
Hence
= 2IL
but
L, then
d B i.
IL
<
- 1
This contradicts our hypothesis,
and the proof is complete.
In the concrete matrix version of the problem, it should
be noted that no matrix operations are necessary until the
end of each cycle (adding an element
y).
The labelling is
done entirely by scanning the nonzero elements of each column.
After the new bases
Bi,B2,...,B'
row operations on each
Mi
have been found, one performs
to put it in canonical form with
-10-
respect to
but it is not necessary to do this sooner.
B!,
B2 = {x2 ,x3 }, and
and
B 1 = {x3,x 4 ,x 5 }
For example, in the picture below, if
y = xl, the circles and arrows illustrate
the relations
1
)(1
-
_I
,
XI
h- -
X,
-
X
O O
C
K---
-
I
O
0 0
WAA
B
2 B1
B2
3
-
C
O
o
o
0
O
M%~
I
j·
---
--
_
Fig. 1
(In fact, this is all the labelling which takes place).
According to the algorithm, we construct new bases
i =
(B1 - x 4 )
%
x 2 = {x 2,x ,x
3
5}
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-11-
Q
= (B2 - x 3 - x 2)
B
and
x4
u
x 1 = {XlX 4}
which provide a complete partition of
X.
A variation on the Edmonds-Fulkerson theorem which can
be proved by similar methods is the matroid intersection theorem
If G 1 (X) and G 2 (X) are two geometries defined
(due to Edmonds):
on the same set
of size
subset
X, then there exists a
which is independent in both
k
and only if
k
< r 1 (A) +
G1
for all
r 2 (X - A)
X
S
G2
and
A
if
. X.
The
connection between matroid intersection and matroid partition
is well known, and a labelling algorithm similar to the one
given above can be constructed.
Such an algorithm has been
described by Lawler [1]. (See also Edmonds [5],[6])
3.
Multiple
xchange Theorems
The following theorem was proved by Greene [8]
(and
independently by Brylawski [2]).
Theorem 3.1:
G(X), and let
such that
S
Let
A
c
(S-A)
T
and
S.
B
be bases of a combinatorial geometry
Then there exists a subset
and
A
(T-B)
B c
T
are both bases.
If S is a singleton, it is not difficult (see [2], [3]) to show this. For
matrices it can be proved immediately by assuming that
a coordinate basis.
The columns of
T
S
is
are represented by a
tonsingular matrix and the result is equivalent to the following:
-12Theorem 3.2
Let M be a nonsingular matrix, whose rows have been
partitioned into two parts
A
and
A'.
possible to permute the columns of
principal minors corresponding to
M
Then it is always
in such a way that the
and
A
A'
are nonzero.
This follows easily from the Laplace expansion theorem for
determinants, but the question of how to carry out the exchange
is much less obvious.
Greene's original proof provided an ef-
ficient but unattractive algorithm.
However, it is much more
convenient to observe that the multiple exchange property is a
trivial consequence of the Edmmnds-Fulkerson theorem.
Hence
an elementary algorithm is easily obtained.
To see this, consider the geometries
G 2 (T)
G/S-A
defined on
That is, we define rank
T
y "factoring out"
rl(U) = r(U
.
r 2(U)
j
= r(U
T
equivalent to partitioning
G1
and
A
(S-A))
S-A.
- r(S-A).
for a subset of
into sets
G 2, respectively.
IlI
U
and
A) - r(A)
B1
and
r l (U) + r 2 (U)
c
T.
But
rl(U) + r 2 (U) = r(U
A)
+ r(U
(S-A))
B2
T
-
SI
is
which
According to the
theorem, this can be done provided that
for every subset
A
and
unctions
It is easy to see that exchanging
are bases in
G 1 (T) = G/A
-13-
=
r(U
A) + r(U
by the submodular law.
But
r((U
A)
(U
)
A J(S-A))
(S-A)))
(U
> r((UvA)
(S-A)) - r(U
(S-A))) = r(U)
=
lUI
and this completes the proof.
In order to apply the Edmonds-Bulkerson algorithm, it
Remark:
is not necessary to compute the factor geometries
G/A and G/S-A.
The algorithm can be applied directly, provided that we start
B1
with bases
A
and
B 2 v (S-A),
B 1 CT,
ify step (1) by requiring that elements of
S
B 2 C.T, and modare never labelled.
The multi-part partition theorem in fact proves a stronger
fesult:
Let S and T be bases of G(X) and let
Theorem 3.3
nI =
{SlS
2 ,...
a partition
,S
H' = {T ,...,Tk}
that, for each
basis of
Proof:
S.
be a partition of
of
Then there exists
with the property
T
i = 1,2,...,k, the set
(S-S.) ) T.
is a
G(X).
To extend the argument used to prove the multiple ex-
change theorem we need the following extended submodular inequality (easily proved by induction, using the ordinary submodular
law):
I
-P1' 2''Pk
if
r(i)
>
r(
are subsets of any geometry, then
Pi) + r(PlU
2 Pi) + r(P2% n
+ ... + r(Pkl
U
Pk ) -
Pi)
-14-
To prove the theorem, let
A
C
T, then
ri(A) = r(A
k
A
=
U (S-Si)
i+l Pj)
r(tk Pi )
= JAI .
for every subset
T
(S-Si))
Gi
Ti V (S-Si)
If
ISI
(k-l)ISI.
in the above inequality.
ISI
for each
Hence
Zr i(A)
=
A
C
T.
for each
i.
i = 1,...,k-l, and
> IAI +
(k-l)ISI - (k-l)ISI = IAI,
Ti
such that
Ti
is indepen-
It is easy to show that this implies
is a basis in
G
for each
i.
is taken to be the trivial partition of
parts, we obtain the following result of
Theorem 3.4
Then
By the Edmonds-Fulkerson theorem,
can be partitioned into sets
dent in
-
i=l
n
r(Pi
If
(S-Si)) - IS-Sil, so that
r(A
I
i=l
Pi
i = 1,...,k.
k
ri(A) =
Let
G i = T/S-S i,
S
into
rualdi [3]:
If S and T are bases of G(X), there exists a one-
to-one correspondence
is a basis for all
x
0:
S - T
such that
(S-x)
(x)
S.
There are elementary examples which show that the last
two results are replacement theorems rather than exchange
theorems.
to have
That is, for example, it is not always possible
(S-x)
bases for all
(x)
x
S.
and
(T-4(x))
(See [31.
x
simultaneously
Dilworth [41 obtained similar
results in a related but somewhat more special case.)
It is interesting to note that the
dmonds-Fulkerson
partition theorem proves a result which is apparently stronger
The referee has pointed out that Theorem 3.3 can be derived
directly from Theorem 3.1 by an induction argument.
-15-
This is most clearly
than the multiple exchange theorem.
seen by examining the analog of Brualdi's theorem when one
what conditions, if
size ISI,
S
under
We ask:
of the sets is not required to be a basis.
is a basis and T is an arbitrary set of
does there exist an injective map
XS-x) J a (x) is a basis for each
x
S.
S + T
:
If
T
such that
is represented
by an arbitrary square matrix, the Edmonds-Fulkerson theorem in this
case gives necessary and sufficient conditions for some term
in the determinant expansion of
conditions are equivalent
conditions" of
T
to be nonzero.
to the well-known
(These
"matching
P. Hiall [9], as can be easily verified,)
Brualdi's theorem, on the other hand, gives only a sufficient
condition:
that the columns of
T
be independent.
In an
analogous way, the 2-part case of the Edmonds-Fulkerson
theorem gives a result which is apparently stronger than'B;-reen's multiple exchange property.
We remark that, when applied to Brualdi's Theorem,
the algorithm which we describe in section 2 is essentially
equivalent to the so-called "Hungarian method" - or "alternating
chain" method - for finding a matching in a bipartite graph.
4.
Sequential
xchange Properties
In this section, we consider the question:
can a
multiple exchange be carried out by a series of single
exchanges?
Here we mean exchange rather than replacement:
-16If
x
S
and
y
T,
a pair of pivots
a single exchange of
x--
y, y--+
S
T
pivot
x --
y
or
y--4
x.
x.
x
for
y
is
A replacement is a single
There are five questions which
S
.T
one might reasonably ask:
Question 1:
If
A
C
S
can be exchanged for
IAI
always possible to do this with
If
exchange
for some
B
C
If
S
can be exchanged for
A
T
C
T, is it
single exchanges?
Question 2;
A = {al,...,ak}
B
is it always possible to
by exchanging
al,a ,...ak
2
in order?
Question 3:
A
C
B
C
T, is
there always some set of single exchanges which carries this out?
Question 4:
a
a
such that
If
A
(1) ,_2_ (2 )"..
Question 5:
A = {al,...,ak}, is there always a permutation
B
can be exchanged for some
,a
Is it
(k)
(k) -
by exchanging
in order?
le to exchange
A
for some
by
some sequence of exchanges?
In this paper, we will partially answer these questions
as follows:
(i) The answer to questions 1 and 2 is no.
(ii)
The answer to question 4 is yes if
k = 2.
-17-
Conjecture:
Questions 3 and 4 (and hence 5) can be answered
affirmatively
for all k.
First, the counterexamples:
M
x1
x2
x3
x4
x5
x6
1
0
0
1
0
1
0
1
0
0
1
1
0
0
1
1
Counterexample 1:
then
let
{XlX2}
If
1
1
S = {x1 ,X 2 ,X 3 }
can be exchanged for
be the matrix
.
and
T = {x4 ,x 5 ,x 6 },
{x4 ,x 5 }
but it is not
possible to achieve this by two single exchanges.
Counterexample 2:
{xl,x3 }
x 1 4- x4 .
Let
S
and
can be exchanged for
T
be as above.
{x4,X 5 }
via
Then
x 3 4- x5,
However, it is not possible to exchange
for anything by switching
x1
first and then
{xl,X 3 }
x 3.
We now prove two lerunas in order to affirm question 4
when
k = 2;
-18-
Lemma 4.1:
Suppose that
S
and
are bases of a combinatorial
T
geometry, and suppose that there exists a closed alternating
chain of pivots
X1 '*
Y1 -
S
X2 -
T
Y2
..
Yn
S
xn+l
-
= X1
T
(Here we assume that the x's are in
T
and the y's are in
S).
If this cycle is minimal, in the sense that it contains no
x. -j- yj, i
i or Yi
* xj, i
j-l, then
S
T
{X...,x
. } can be exchanged for {ylY2...,y }.
chords
Proof:
This is a special case of the lemma on sequential
pivots described in section 2.
Next, we have the following lemma, which should not be
confused with the (false) assertion in Question 1:
Lemma 4.2:
B
S
Suppose that
T, with
=
AI
[BI
=
and
k.
If
T
A
are bases and
A
Proof:
B,
2k
(or pivots).
Consider the directed graph whose vertices are the
elements of
a --
S,
can be exchanged for
it is possible to carry out this exchange by means of
replacements
£
A
b, b' -
T
to some
b
B, and whose edges are given by the symbols
a'.
S
B
First observe that every
a
A
is connected
by an edge
on
a--4 b, since otherwise a depends
T
T - B, which is impossible since A can be exchanged for
B.
Similarly, eac
b,
B
is connected to some
a
A.
Hence
-19-
there exist directed cycles, and we choose one which is minimal.
By the previous lemma, this permits us to exchange some subset
A0
c
B
for some subset
where
k
=
A0 1 =
B 0 1.
B1
C- B, using
2k 0
replacements,
Now repeat the process for
A - A0,
B - B0 , and so forth until the exchange is complete.
Remark:
It is possible to use the previous two lemmas to construct
a labelling algorithm for multiple exchange directly.
However,
it is entirely equivalent to the one previously described
o
we omit the details.
If our conjecture is true, the
2k
pivots described in
the previous lemma can be arranged so that each successive
pair
x-+ y ,y-
x
is an exchange.
is always the case if
Theorem 4.3:
Let
S
Next we show that this
k = 2.
and
Then, after relabelling
T
xl
be bases, and let
and
x
{XlX
2 }
C
S.
if necessary, it is
possible to find a sequence of exchanges
X1-
T
x2
for some
Proof:
,y
T1.
Suppose that
always possible).
If
- Y2
T'
(here
x1
Y1
S' =
X1
S
$'
2
(S-x1 )
y1 '
has been exchanged for
x2
we are done, so assume that
T' = (T-y )
yl (as is
can now be exchanged for some
x2
can be exchanged only for
Y2
xl.
x
)
-20-
S" =
This implies that
are both bases.
(S-x2 )
Yl
and
T',
{y1 'X1 }
(T-yl)
On the other hand, we know that
can be exchanged for something, say
and
T"
{y1 ,x 2 }
{Y2 'y3 }.
can be exchanged for
can be exchanged for
S'
x1 ,x 2 1
Hence, in
{y2 ,Y
{y2 ,y 3 } in
x2
3
}
S'
Similarly,
and
T".
By
the previous lemma, each of these exchanges can be carried
out by four pivots, which we represent by the following diagrams:
Sh
'T'
'
s'
S"
T'*
S"
We can assume that the diagrams have this form, since any chords
would permit a sequential exchange immediately, and the possibility
-
j
,
X
a-
i
_
for the second diagram is excluded by the fact that the arc
X1 a 19
Y3
must be present.
(This follows from the existence of
T"
arcs
xlT"
replacing
x2
x2
and
x2
T'
by
x1
in
Y 3 , since
T".)
Hence
- S-x 1.
But then
Y2
depends on both
Y2
depends on
Y 2 ----
S'-y
S'
1
y1
S-x2
nor
and
Y 2 '-"
S"
S"-y 1
Y1
S-xl-x 2, which contradicts the
fact that {Y2 ,Y 3 } can be exchanged for {xl,
proof.
is the result of
From the fact that both chains
are chordless, we infer that neither
occurs.
T'
2 }.
This completes the
-21-
Note:
After submitting this manuscript, the authors learned
that D. E.
Knuth had independently discovered the same proof
of the Edmonds-Fulkerson partition theorem
tioning") Stanford Technical Report Stan
Knuth also employs an
("Matroid PartiCS-73-342, March 1973).
"arrow" notation which is similar to ours.
-22-
References
1.
Bondy, A, "Pancyclic Graphs II", Proc. Second Louisiana
Conference on Combinatorics, Graph Theory and Computing,
Baton Rouge, 1971.
2.
Brylawski, T., "Some Properties of Basic Families of Subsets",
Disc. Math. 6 (1973), pp.333-341.
3.
Brualdi, R., "Comments on Bases in Independence Structures",
Bull. Australian Math. Soc. 1 (1969), pp.166-167.
4.
Dilworth, R. P., "Note on the Kurosch-Ore Theorem", Bulletin
of the AMS 52 (1946), pp. 6 5 9 - 6 6 3 .
5.
Edmonds, J.,
6.
Edmonds, J., "Submodular Functions, Matroids, and Certain Polyhedra",
"Matroid Intersection",
(preprint).
Conmbinatorial Structures and their Applications (Proc.
Calgary Conference), Gordon and Breach, (197,), pp.69-8 7 .
7.
Edmonds, J., Fulkerson, D. R., "Transversals and Matroid
Partition", J. Res. Nat. Bur. Standards, 69B (3) (1965)
pp.147-153.
8.
Greene, C.,
"A Multiple Exchange Property for Bases", Proc.
AMS 39 (1973), pp. 45-50.
9.
Hall, P., "On Representatives of Subsets", J. Lond.
(1935), pp. 26-30,
ath. Soc., 10
10. Holtzmann, C. A.,
arary, F.,"On the Tree Graph;of a Matroid"
SIAM J. Appl. Math. 22 (1972) pp.187-193.
11. Lawler, E., "Optimal Matroid Intersections", Combinatorial
Structures and Their Applications (Proc. Calgary Conference)
Gordon and Breach (1970) p. 2 3 3 .
12. Maurer, S. B., "Matroid Basis Graphs", I,II, Journal Comb.
Theory (B) 14 (1973) pp.216-240, 15 (1973) pp.121-145.
13. Maurer, S. B., "Basis Graphs of Pregeometries", Bull. AMS 79
(1973) pp.783-786.
14. Rota, G.-C., "Combinatorial Theory", notes, Bowdoin College,
Brunswick, Me. (1971).
15.
Rota, G.-C., "Combinatorial Theory, Old and New", Proc. Int.
Congress, Nice, 1970, Vol. 3, Gauthier-Villars, Paris
(1971), pp.229-234.
-23-
16.
White, N., "Brackets and Combinatorial Geometries', Thesis,
Harvard University, Cambridge, Mass., 1971.
17.
Whitely, W., "Logic and Invariant Theory", Thesis, MIT,
Cambridge, Mass., 1971.
18.
Whitney, H., "On the Abstract Properties of Linear Dependences",
Amer. J. Math. 57 (1935), pp. 50 9 - 53 3 .
-24-
NV
1
2
I
x
o
-
re
x
ct
x
o
o o
-I
o
--
I
0
0001
0
O0
x
I
111.
I
i~~~'
-25-
Symbols appearing in text:
set-membership (epsilon)
set-non-membership
C
set-inclusion
W
set-union (small)
el
set-intersection (small)
set-intersection (large)
U
set-union (large)
arrow
[Note: all arrows are intended to be
approximately the same length]