0
6
5
1+
0 6 5
7 1 0
8 7 2
9 8 7
1 9 8
3 2 9
5 1+ 3
2 3 1+
1+ 5 6
6 0 1
7
1
0
6
8
7
2
1
9
8
7
3
1+ 9 8 7 1 2
6 5 9 8 2 3
1 0 6 9 3 1+
3 2 1 0 1+ 5
7 1+ 3 2 5 6
8 7 5 1+ 6 0
9 8 7 6 0 1
5 6 0 1 7 8
0 1 2 3 8 9
-2 3 1+ 5 9 7
1
9
8
7
3 5 2 Ij 6
2 Ij 3 5 0
9 3 1+ 6 1
8 9 5 0 2
8 6 1 3
3
7 0 2 Ij
1+
6 1 3 5
5
0 7 8 9
6
1 9 7 8
0
2 8 9 7
1
COM BIN A TOR 1 A L
MATHEMATICS
YEA R
2
9
7
8
Febl'Ua:l'Y 1969 - June 19'10
MATCHING THEOREMS FOR COMBINATORIAL GEOMETRIES
by
MARTIN AIGNER
and
T. A. Dowling
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 600.12
AUGUST 1969
Research was partially supported by the U.S. Air Force under Grant No.
AFOSR-68-l406 and the National Science Foundation under Grant No. GU-2059.
Matching Theorems for Combinatorial Geometries
by
Martin Aigner
and
T. A. Dowling
Department of Statistics
University of North Carolina at Chapel Hill
1.
Let
and
G(S)
triple
B
G(T)
be combinatorial geometries [2] on sets
T, respectively, and let
points of
and
and
G(S)
and G(T).
(A, B, f),
G(T),
such that
A and
respectively, and
E
R
~
SxT
f
B
G(S)
into
G(T)
is a
are independent sets in
G(S)
is a one-one function from
R for all
S
be a binary relation between the
A matching from
where
(a, f(a))
INTRODUCTION
a
E
A onto
A.
The present paper presents a characterization of matchings of maximum cardinality, a max-min theorem, and a number of related results.
In the case where both
G(S)
and
G(T)
are free geometries, Theorem 4
reduces to a characterization of maximum matchings associated with the
"Hungarian method" as introduced by Egervary and Kuhn (see [1]),
Theorem 5 to the Konig-Egervary Theorem, Theorem 6 to a theorem of Ore
[5], and
Corollar~
6.1 to the classical Marriage Theorem.
corollary for the case when
G(S)
is free and
obtained by Rado [6J (see also Crapo-Rota [2J).
G(T)
The latter
arbitrary was first
Theorem 7 is known (see
[2J, [3J, [4J), but the proof, based on earlier results in the paper,
is new.
Research was partially supported by the U.S. Air Force under Grant No.
AFOSR-68-l406 and the National Science Foundation under Grant No. GU-2059.
2
2.
DEFINITIONS, NOTATION, AND TERMINOLOGY
For completeness, several definitions and results on combinatorial
geometries from Crapo-Rota [2] are included in this section.
A closure relation on a set
all subsets
A
~
S,
A
~
A,
(2.2)
A
~
B implies
A, B
a closure space.
A
A
defined for
A set endowed with a closure relation is
A
~
is closed if and only if
S
A closure relation on a set
Any subset
that
~
B,
~
S.
of
A subset
(2.3)
A
satisfying
(2.1)
for all subsets
is a function
S
A
O
A
S
= A.
has finite basis if and only if
S
c
A
has a finite subset
A
O
c
A
such
= A.
A closure relation satisfies the exchange property if and only if
For any elements
A
c
S,
a
E
Aub,
a, b E S,
and for any subset
(2.4)
a
~
A
implies
A pregeometry (matroid)
a set
S
property.
G(S)
b
E
is any closure space consisting of
and a closure relation with finite basis and the exchange
A pregeometry
G(S)
is a combinatorial geometry if and only
if
(2.5)
Aua.
¢,
a
=
a
for all
a E S.
3
Associated with every pregeometry
etry
G(SO)
on the set
So
G(S)
on a set
S
is a unique geom-
of equivalence classes of
S
under the
equivalence relation
a
~
b
i f and only i f
a
=b
•
We shall confine our attention to geometries with no
loss of generality;
the results may easily be extended to pregeometries.
The cardinality of a set
G(S) ,
A
a =
all minimal subsets
A
S
A
O
will be denoted
of any subset
A.
A set
A
r
of
G(S)
~
r(AuB) + r(AnB)
S.
satisfying
A
has closure
r(A)
r(A)
v (A) ,
A.
The rank
satisfies the semi-modular ineguality
~
A combinatorial geometry
A
S
is independent if and only if
i.e., if and only if no proper subset of
(2.6)
~
In a geometry
have the same cardinality, which is defined as the rank
of the set
function
A
v(S).
r(A) + r(B).
G(S)
is free if
In this case, every subset of
S
A
=A
for all subsets
is both independent and closed.
The concept of independence is thus non-trivial only with regard to
"multisets" on "lists", in which an element may occur more than once.
A multiset in a free geometry is independent if and only if all its
elements are distinct, i.e., if and only if it is a set.
The following definitions and notation are introduced in the present
paper.
For a further theory of combinatorial geometries, the reader is
referred to [2].
Let
G(S), G(T)
tively, and let
and
T.
R
~
be combinatorial geometries on sets
SxT
respec-
be a binary relation between the points of
The system consisting of
(G(S), G(T), R).
S, T,
G(S), G(T), and
R
S
will be denoted
We shall denote the rank functions of both
G(S), G(T)
4
by
B
and the closure relations by
r
~
T.
(a,b)
For
R
E
A
~
s,
for some
A matching in
R(A)
a
a
A,
E
and
A matching
tively.
B
A,
-+
B,
where
T
A
onto
(A, B, f),
B such that
(a, f(a»
~
S,
such that
(A, B, f)
The common cardinality of
R
f
for
respec-
is characterized by its edge set
{ (a, f (a»:
=
where
E
G(S) , G(T),
a
A},
E
and we formally identify these two concepts by writing
M.
E
is a triple
are independent sets in
M
matching
b
A
A.
E
(G(S), G(T), R)
A, B
-+
denotes the set of points
is a one-one function from
all
A
A, B, M is called the size
M = (A, B, f).
v(M)
of the
We shall be interested in matchings of maximum size in
(G(S), G(T), R).
A support of
G(S), G(T) ,
of
c
d
E
A(C, D)
number
D holds.
S
to
(G(S), G(T), R)
T,
(c, d)
(C, D)
E
of closed sets in
R implies at least one
The order of a support
(C, D)
is the
r(C) + reD).
Note that if both
system
is a pair
respectively, such that
C,
E
(G(S), G(T), R)
G(S)
and
G(T)
are free geometries, then the
is, apart from the orientation of the edges from
a bipartite graph, and the above definitions of a matching
and a support reduce to the usual ones for this case.
The following
definition uses the exchange property to generalize a notion associated
with the "Hungarian method" for finding a maximum matching in a bipartite
graph.
Let
(2.7)
M
(A, B, f)
be a matching in
(G(S), G(T), R).
A sequence
5
of
2n + 1
to
M
distinct pairs
(n::> 0)
if and only if
(a ~ , b~+1)
l
(2.8)
(a. , b. )
N,
(2.9)
a'
f
S-A,
b~+1
(2.10 )
a~
E
A,
l
0
l
l
b~
E
l
for
1
~
i
~
B,
a~
l
b'
i
l
= a
i'
~
(A
R-N,
T-B,
E
i-l
i
-
U
a. ) u
J
j=l
~
E
U
a~ ,
j=l
J
i-l
i
U b .) u U b ~ ,
j=l J
j=l J
(B -
n.
Note that if both
a~
is an augmenting chain with respect
b' = b
i
i
G(S), G(T)
1 s i ~ n,
for
are free geometries, (2.10) implies
so that the sequence represents an
ordinary augmenting chain in the bipartite graph.
3.
THEOREM 1.
matching
MATCHING THEOREMS
If there exists an augmenting chain with respect to a
M = (A, B, f)
in
(G(S), G(T), R),
then
M
is not of maximum
size.
PROOF.
Let the augmenting chain be given by (2.7) and define
P
{(a.,b.):
l
l
l~i~n},
A s traigh tforward inductive argument using (2.10) and the exchange
property shows that
6
i
i
(A -
U a. ) u U a~
j=1 J
j=1 J
(B -
U b.) u U b~
j=1 J
j=1 J
and
i
i
are independent sets with closures
1
~
i
~
n.
(3.1)
A, B,
respectively, for all
i,
Thus by (2.9),
(A -
n
n
U a.) u U a'
j
j=1 J
j=O
and
n
(3.2)
(B -
n+1
U b.) u U b'
j=1 j
j=1 J
are independent sets of cardinality
M'
(3.3)
The edges of
(M - P) u P'
define a one-one function
matching in
v(M) + 1.
f'
of (3.1) onto (3.2), so
(G(S), G(T), R),
v(M') = v(M) + 1.
and
M'
Thus
is a
M is not
of maximum size.
THEORE}l 2.
If
(G(S), G(T), R),
M = (A,B,f)
is a matching and
then
v(M)
PROOF.
(C,D)
By definition,
R(S-C)
v (M)
~
~
D.
A(C,D).
Therefore
v (A)
=
v(AnC) + v(An(S-C))
v(AnC) + v(f(An(S-C)))
is a support in
7
= r(AnC) + r(f(An(S-C»)
~
r(AnC) + r(R(An(S-C»)
~
r(C) + r(R(S-C»
~
r(C) + reD)
= A(e, D) .
THEOREM 3.
M = A,B,f)
If
is a matching in
(G(S), G(T), R)
does not exist an augmenting chain with respect to
a support
(A-Am' f(Am»,
where
M,
and there
then there exists
Am -c A.
The proof of Theorem 3 is constructive.
We require several lemmas
before proceeding with the main proof.
LEMMA 1.
If
B , B
i
2
are independent sets in
G(T),
and
B u B
i
2
is
independent, then
=
PROOF.
B nB
i
r
2
of
Clearly
~
B nB •
i
2
B nB
i
2
~
B nB ,
2
i
and since the latter is a closed set,
By the semimodular inequality (2.6) for the rank function
G(T),
r(B nB )
2
i
~
r(B ) + r(B ) - r(B uB )
i
2
i
2
r(B ) + r(B ) - r(B uB 2 )
2
i
i
and the lemma follows.
=
v(B ) + v(B ) - v(B UB )
i
2
2
i
=
v(B nB )
i
2
=
r(B nB ),
i
2
8
LEMMA 2.
Let
B be an independent set in
G(T)
and suppose
D
~
B.
~
B ,
2
Then the set
B
1
{b
is the unique minimal subset of
PROOF.
then there exists
b
E
B
B-b}
B such that
such that
1
~
D.
B whose closure contains
B be any subset of
2
Let
D
B:
E
B
B-b.
c
2 -
D
~
B ·
2
But then
If
B
1
D
~
B-b,
1
~
i
a
contradiction.
Using Lemma 1 and the definition of
b
I
E
b
I
E
B-B
1
n
3.
Suppose
B
we have
1
n
D
LE~lliA
B ,
(B-b ')
B-B
1
is an independent set in
c
•••
is an increasing sequence of subsets of
B.
c
b.1
(ii)
b~
E
Let
B.1 - B.1- l'
B. - B-b ..
E
111
Then
b
for
1
~
i
~
n.
l
i
~
(B -
i
U b.)
1
j=1
U
and
B
n
be points satisfying
(i)
G(T)
i-1
U b~
. 1 1
J=
b., b ~ ,
1
1
~
n,
9
PROOF.
We first show that
B'
i
U
i
1
is an independent set with closure
b 1'
E
B1
B~
implies
result true for
=
i-l~
B
1
i
b.) u U b I
j
j=l J
j=l
(B. -
Now by (ii)
B .•
1
by the exchange property.
let
i-1
(B. 1
i-1
U b.) u U b '
j
j=l J
j=l
(B.1 - B.1- 1) u B~1- 1
Then
C.
1
(B. - B. 1) u B ~ 1
1
(B. 1
1-
1-
B. 1) u
1-
B~1- 1
(B. - B. 1) u 13. 1
1
1-
= (B.1 - B.1- 1)
B.
1
1-
u B.1- 1
~
and by a similar argument
C. - b.
1
1
(B.1 - B.1- 1 - b.)
u B~ 1
1
1B. - b .•
1
1
so
Assuming the
10
It follows now from (ii) and the exchange property that
B.
C.
1
1
(c. - b.) u b~
111
so by induction
(B -
B'
has closure
i
i-1
i
U b.) u U b'
1
i
j=1
j=1
B.
1
1 :::; i
for
:::;
n.
Therefore
(B - B.1- 1 - b.)
u B~ 1
1
1B - b.,
1
and the lemma follows by (ii).
LEMMA 4.
Suppose
A
is an independent set in
is an increasing sequence of subsets of
A.
G(S)
Let
be points satisfying
(i)
a.1
E
A.1 - A.1- l '
(ii)
a~
E
A-A. 1 - A-a ..
1
1-
1
Then
a' ~ (A i
for
1:::; i :::; n.
i-1
i
U a.) u U a'
j
j=1
j=1 J
and
a~ ,
1
1 :::; i
:::; n,
11
PROOF.
for
al ~ A-a ,
1
By (ii),
i -1.
so assume inductively that the lemma holds
Then
i-l
i-l
U a.) u U a'
j
j=1 J
j=1
(A -
is an independent set, and thus so also is
(A~ 1 -
Since
i'
A~
l - l'
a.) u (A - A. 1)
l
l-
l-
a
A*
i-1
it follows from Lemma 1 that
A-A. 1-a . .
l-
Hence if
al~ E A~ 1-a.,
l-
l
l
then by (ii),
a~
A-A. i-a. ~ A-a.,
ll
l
E
l
a contradiction.
PROOF OF THEOREM 3.
Let
C
0
=
S - A.
no augmenting chain with respect to
subset of
B,
C
1
we define
B.
E
B.l - l'
o)
= (A,B,£).
=S -
A-A .
Let
Thus by Lemma 2,
A. = f
-1
l
B. 1
l-
(A.),
l
~
B.
l
C
i
and so
B
Thus
process terminates.
i
~
m,
where
B
O
= ¢.
be the minimal
R(C ) -c B1 .
O
A-A .•
l
A. 1
l-
R(C) nB c B,
m
m
but
c
l
m
Since
A.,
l
A., B d
strictly increasing up to and including some index
~
B
1
such that
= S -
It is, moreover, clear that each of the sequences
o
since there is
::: B
In general, having constructing
1
as the minimal subset of
l
and set
b
M
R(C
defined according to Lemma 2, such that
Then let
c.l - l '
Then
l
is
C.
l
after which the
R(C.) n B ~ B.
l
l
for
12
We shall show that
otherwise, let
5
R(C )
n
B.
n
R(C.)
l
B
c
-
for all
be the smallest index,
i,
0
~
n
0
~
~
m,
~
i
m.
Assuming
such that
We obtain a contradiction by showing that this assumption
implies the existence of an augmenting chain with respect to
Since
n
(a I
an edge
5
R(C)
b
n'
I
n+1
B,
but
R(C.)
al
n
al
n
exists
a
for all
bn
A-a
E
n
a _
n 1
b'
n
E
n
a
A - ,
n 1
such that
a lEe
C
n-l
n-l- n-2
0
~
i
n,
<
a l ~ A-a.
n
n
and hence
al
n-1
Since
and
al
n
then
E
R(C
a
n
B
A-A
Since
al
n
A -A l'
n n-
E
by Lemma 1, so there
n
C
n-1'
b'
n
) c B
n-1
E
A-A
E
Let
and Lemma 2,
n
n-2
E
B -B-b .
n
n
b
n
al
n
l'
E
= f(a n ),
A-a
n-1
then
Thus
b n' ~ ~1'
n-
and
by hypothesis, it follows that
(a I
b I) 6; M.
n-l' n
and arrive finally at a sequence
(3.4)
where
(3.5)
(3.6)
a~ E S-A,
(3.7)
a.
E
A.-A.
l'
ll
a~
E
b.
E
B.-B.
l'
ll
b~
l
E
l
n-
there exists an edge
We may then continue this argument starting with
l
there exists
n-1
By definition of
B -B l '
n n-
for
T-B ,
E
A,
n
E
n
such that
E
Bn -B n- l'
E
( a I l ' b I)
nn
so
A
n
E
C -C
E
for all
n
B
such that
)
b~+1
If
c
l
M.
b~+l E T-B,
l
A-A·
l- 1
-A-a.,
l
---
B .-B-b. ,
l
l
a~_l
E
C - -C - ,
n 1 n 2
13
for
1
i
$
n.
$
Now (3.5) and (3.6) are simply (2.8) and (2.9), and by
Lemmas 3 and 4, (3.7) implies (2.10).
It follows that the sequence (3.4)
in reverse order represents an augmenting chain, which contradicts the
hypothesis of Theorem 3.
Thus
R(C _ ):: B
i
i 1
S - A-A
C
m'
m
the pair
for
1
$
i
$
(A-Am' B )
m
m,
and
R(C) c B .
m - m
Since
is a support, and the proof of
Theorem 3 is complete.
THEOREM 4.
A matching
M = (A,B,f)
in
(G(S), G(T), R)
is of maximum
size if and only if there does not exist an augmenting chain with respect
to
M.
PROOF.
The necessity of the condition follows by Theorem 1.
If there
does not exist an augmenting chain, then the support given by Theorem 3
has order equal to the size of
plies that
If
M = (A,B,f)
there exists a matching
is a matching not of maximum size,
M' = (A'ua,
B'ub, f')
such that
A' = A ,
B.
PROOF.
M,
which together with Theorem 2 im-
M is of maximum size.
COROLLARY 4.1.
B' =
M,
By Theorem 4, there exists an augmenting chain with respect to
and the required matching
M'
is constructed as in the proof of
Theorem 1.
THEOREM 5.
The maximum size of a matching in
to the minimum order of a support.
(G(S), G(T), R)
is equal
14
PROOF.
If
M = (A,B,f)
is a matching of maximum size, then by Theorem
4 there does not exist an augmenting chain with respect to
of order
v(M)
M.
A support
therefore exists by Theorem 3, and this support is nec-
essarily of minimum order by Theorem 2.
Following Ore [5] for the case of a bipartite graph, we define the
deficiency
0S(A)
of a subset
A
~
S
by
r(S) - r(S-A) - r(R(A)),
and let
Note that
THEOREM 6.
Os
~
0
0S(~)
since
In the system
(G(S), G(T), R),
max
v(M)
M matching
PROOF.
form
= o.
min
A (C,D)
(C,D) support
r(S) -
Os .
Note that every support of minimum order is necessarily of the
(C, D),
where
D
=
R(S-C).
r(S) -
Now
max [r(S) - r(S-A) - r(R(A))]
A~S
min [r(S-A) + r(R(A))]
A~S
min [r(A) + r(R(S-A))],
A~S
and the latter minimum is clearly attained when
A is a closed set.
15
COROLLARY 6.1.
There exists a matching of size
r(S)
in
(G(S) ,G(T) ,R)
i f and only i f
r(S) - r(S-A)
for every subset
THEOREM 7.
A
s
S.
C
(See also [2], [3], [4],)
in the system
r(R(A»
(G(S), G(T), R),
there exists a matching
If the geometry
then the subsets
(S', T', f)
for some
G(S)
S'
T'
of
and
S
f,
is free
for which
are the in-
dependent sets of a pregeometry, the transversal pregeometry on
PROOF.
Let
I
be the family of subsets
exists a matching
S'
C
-
S,
let
oS'
= O.
for some
of
T'
and
be the free subgeometry on
G(S')
6 to the system
if
(S', T', f)
S'
(G(S'), G(T) , R n (S'xT»,
Equivalently,
S'
E
I
S
S.
for which there
f.
S' .
we have
if and only i f
Given any subset
Applying Theorem
S'
E
I
i f and only
v (S') s v(S')-oS"
The theorem will therefore follow from Proposition 7.3 of [2 ] if the
function
r*(S')
is increasing and semimodular.
for
v(S') - oS'
We proceed to establish these properties
r*.
Let
S' ~ S
and
A
c
S'.
Since
G(S')
is free, the deficiency
v (A) - r (R(A»
is independent of
S'
,
so we may omit the subscript.
max
A
c
S'
°(A).
Then
16
Given
A , A c S',
1
2
it follows from the relations
and the semimodular inequality (2.6) for the rank function
r
of
G(T)
that
(3.8)
Let
F(S')
be the family of subsets
Then by (3.8),
F(S')
A
of
S'
for which
Suppose that
a
a
E
S'-~"
As,.
E
<5
=
oS'.
is closed under the operation of intersection,
and therefore contains a minimal set, which we denote by
Clearly if
o(A)
then
Since
AS'
AS'-a
= AS'
and
AS'.
0S'_a
=
oS'·
0S'-a ~ 0S,-l.
is minimal,
But
v(AS ' - a) - r(R(A S ' - a»
(AS' - a)
(3.9)
so
oS' -a
0S,-l.
We have therefore that
r*(s,) - 1,
a
E
S' -AS' ,
r*(S' - a)
{
Thus the function
r*
r*(S'),
is not only increasing, but unit-increasing.
Note from the argument above that if
AS,-a
E
F(S'-a)
and
AS'-a ~ ~,-a.
and
o(A )
S
1
=
a
E
S',
then
We conclude that if
o(A
S
n S1).
2
S1
c
S2 c S,
17
Now let
51' 52
be any two subsets of
A
5 uS
1 2
5,
n (51-52)'
A
n (S2- S1) ,
A
n (S1 nS 2)·
S1 uS 2
Sl u5 2
and let
Then by (3.8) and the above remark we have
Thus
and the proof is complete.
It should be noted that Theorem 7 is false if the geometry
is arbitrary.
G(S)
The function
r*(S')
r(S') - oS'
is unit-increasing, but not semimodular in general, so that Proposition
5.7 of [2] cannot be applied.
For the same reason, Theorem 6 cannot be
18
proved by extending Ore's inductive argument [5] for the case of a bipartite graph to the general case, although this approach works when
G(S)
is free and
G(T)
arbitrary.
ACKNOWLEDGMENT
We would like to express our gratitude to Professor Rota who suggested this work to us during a series of lectures given at the
University of North Carolina in the Spring 1969.
REFERENCES
[1]
C. Berge, The Theory of Graphs and Its Applications,
and Sons, New York, (1962).
[2J
H.H. Crapo and G.C. Rota, On the Foundations of Combinatorial
Theory: Combinatorial Geometries, to appear.
[3]
J. Edmonds and D.R. Fulkerson, Transversals and Matroid Partitions,
J. Res. Nat'l. Bur. Std. 69 B (1965), 147-153.
[4]
L. Mirsky and H. Perfect, Applications of the Notion of Independence to Problems of Combinatorial Analysis,
J. Combinatorial Theory 2 (1967), 327-357.
[5J
O. Ore,
[6J
R. Rado, A Theorem on Independence Relations, Quart. J. Math.
(Oxford) 13 (1942), 83-89.
Graphs and Matching Theorems,
625-639.
John Wiley
Duke Math. J. 22 (1955),