Balanced Matroids
nl
Dalancea ‘ Matroids Tovnds
IBM
Almaden
650 Harry
Milena
FedeF
Center
Research
Road,
Bell
San Jose,
CA
445 South
95120
Abstract
We establish strong expansion properties for the basesexchange graph of balanced matroids; consequently, the set of bases of a balanced matroid can be sampled and approximately counted using rapidly mixing Markov chains. Specific classes for which balance is known to hold include graphic
and regular
matroids.
Morristown,
the set of vectors.
A subclass
lar
will be of interest
Introduction
and
they
matroids:
graphic forth
matroids
denoted
nected
whose vertex-set
S be a finite
ground-set,
tion of subsets of S. Following the set B is said to form matmid
&f(S,
is the collection
operation
the same cardinality
element:
The bases-exchange
has been studied
graph
combinatorial
contexts
related
I?2 = 131\{e ~~}.
properties
for any pair
exchange
property
is that
terminology,
e E B indicating
of bases of a
matroid
of bases J31 and holds:
an ~C l?2 such that ample
standard
for
all
131\{ e}U{~}
of graphic
of G(M).
ple is that maximum
from
the event
e is in the chosen basis. the negative
The cor-
if the inequality
linearly
ex-
and whose bases Another
examis
and whose bases are
independent
< Pr[e] Pr[~]
exists
whose ground-set
matmids,
Pr[e~]
whose ground-set
matmids,
over some field
cardinality
there
is in t3. A main
trees of the graph,
of vectorial
a set of vectors
at random
of S, let e denote
&f (S, l?) is said to satisfy property
relation
l?2 the following
e E l?l
is the set of edges of a given graph are the spanning
that
before in
[12, 20]; here we focus
the mnk of the matroid),
(called
and (ii)
of
of removing
and adding one ground-set
23, and e is an element
if (i) all sets in B have
hence-
by an edge if and only if 132 can be obtained
and let B be a collec-
the collection
23) if and only
M,
by Edmonds
two bases B1 and 132 con-
If B is a basis chosen uniformly Let
All
[28].
131 by the fundamental
on expansion
Preliminaries
of regu-
over every field.
was introduced
with
matroids
is that
of a matroid
gmph
by G(A4),
[10] as the graph
of vectorial
in this paper
are regular
bases of the matroid, from
NJ 07960
are vectorial
The bases-exchange
various 1
Research
Street,
which We introduce the notion of ‘balance” , and say that a rnatroid is balanced if the matroid and all its minors satisfy the property that, for a randomly chosen baais, the presence of an element can only make any other element less likely.
Mihail
Communications
subsets of
holds for all pairs of distinct
elements
ize that negative
is equivalent
Pr[e],
correlation
thus expressing
ence of an element less likely;
the intuitive
studied
before
graphic
matroids
in
fact
~ can ordy make
the negative
correlation
[5, 23].
to Pr[el ~] < that
the pres-
another
element
property
Regular
are known
e, f in S. Real-
and
e
has been
in particular
to be negatively
corre-
lated. *Work
done
primarily
while
the
author
was
at Bellcore.
Here
Permission to copy without fee all or part of this materisl is granted provided that tha copias are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appaar, and notica is given that copying is by permission of the Association for Computing Machinery. To copy otharwisa, or to rapublish, requiras a fee and/or specific permission. 24th ANNUAL ACM STOC - 5/92/VICTORIA, B. C., CANADA @1992 ACM ()-89791 -51 2-719210004/0026 . ..$1.50
we strengthen
lation
to that
is 6alanced
M(S,
B)
itself,
satisfy
minor
the
then
or those this
selecting
operation
if
all
negative
notion
of negative
We its
say that
minors,
correlation
is obtained
operation
that
of selected
26
the
of a matroid
forming and
the
of “balance”:
either
do not.
those
including
M
property.
(A
repeatedly
an element
bases
that
In the case of graphic
corresponds
edges from
by
of choosing
the
to contraction graph.)
corre-
a matroid
pere E S
cent ain matroids, or deletion
e
We first exchange
establish graph
(Theorem
G(M)
both
of G(A4 ), the number
partition
classes is at least
as large
partition
sion properties
had been conjectured
class.
of the to
as the size
Such strong
expan-
by Mihail
[19], in fact for signific=tly
graphs.
M
of edges incident
of the smaller Vazirani
the basesmatroid
1, i.e., for any bipartition
has cutset expansion vertices
3.4) that
of any balanced
wider
correlation
property
negative
correlation
~ implies
and an arbitrary
monotone
between property
and, in view
of the fact
anced matroids, gorithm
e and
an element
e
m on S\ {e},
●
for
their
of self-reducibility an efficient
to approximately
Our results
troid
schemes
we obtain
of any balanced
and
classes of
for two elements
approximation
count
matroid.
of bases of a matroid
The core of our proof here is to show that the
negative
Monte-Carlo
the number
In general,
can be efficiently
for bal-
randomized
al-
of bases
exact
is #P-complete
thus show that
size,
counting
[27].
the set of bases of a ma-
sampled
and approximately
counted
if balance
can be established
for the matroid.
Balance
is known
to hold for graphic
and more gener-
ally regular
matroids;
some counter-examples
to bal-
ance are also known. Pr[em] which
implies
hood
an enforcement
analogous
subgraphs ment
to
(formalized
(this
Previous
encode paths the discrete geometry
neighborfor
as a ‘(ratios finally
implies
for expansion
using
elements
and bound version
certain enfome-
last step was observed
arguments involved
We further
expansion
in [18]), which
condition”
,
of vertex
bipartite
on G(M)
sion for G(M) graphs
< Pr[e] Pr[m]
expanin [18]).
of isoperimetries
from
and the path proof
to
bases-exchange known
significance graph
derives from
ideas, see for example
of the bases-exchange
of the
a sequence of well
[1, 3, 9,14,15,
19, 25],
Consider
the natural
on G(M):
with
random
is the state
If Xt
one half Xt+l
e from if X’
Xt = Xt\
wise Xt+l chain Xt
{e} u {f} = Xt.
uniform import
oracle),
proaching
●
In turn,
reducible
efficient
say, an Most
of G(A4 ) suggests that Xt
and hence possesses the rapid amounts,
roughly,
distribution
[81 ). Therefore,
uniform sampling balanced matroid.
(given,
to the
which
can
it
other-
the Markov
converges
its stationary
on G(M)
efficiently
and that
has large conductance,
for t= poly-log(
= X’,
and,
be used
to Xt
arbitrarily
the natural as an efficient
apclose
random almost
scheme for the set of bases of any
here are of different
Diaconis can yield
combinatorial
uniform populations
sampling yields
that
walk
natural
random
●
c is
The
cardinality
a bound
on total
a technique
O ((n log rn+log
than
(Theorem
5.1).
matroids, of
O((n log rn+log
version
of the
significance
where
the
variation
of
n is the
ground-set,
distance,
In particular,
and conduct~ce
and
we obrandom
E–l )mn2 ) convergence this
arguments
C-* )m2n4)
result
applies
rate to
and significantly imbounds. For graphic
for the nat Ural random
efficient
conductance
rate for the naturzil
algorithmic
graphic and regular matroids proves upon previously known pling
of
[17] we show that
rather
scheme based on the natural
walk with
for self-
symmetry
is as follows: balanced
tain a sampling
to en-
to possess symmetry
convergence
arbitrary the
was Jerrum
type; in fact, matroid
[7] and Mihail
walk.
previously
strong
by extending
m
is
with
as well as for a modified
improvement
For
rank,
only
have been successful
appear
uses paths
a sharper
random this
do not
and Strook
these
any edge
graphs [6, 14, 19]. Our path
Furthermore,
an analysis
matroids, ahnost
graphs
the
and
of using the state-space
arguments
proba-
the ratios
through
The
[14]; such arguments
graphs
from
path congestion
argument
e.g. matching
properties.
Choose
to bound
used here
matchings,
properties, related
over 23 (it is symmetric).
antly, the expansion property
Xt+l
...
edge
of establishing
of paths
graph.
only for combinatorial
t, then
and with
It is easy to see that
distribution
mizing
at time
at random
B then
c
t = 0,1,
as follows:
S uniformly
can be simulated
independence
= Xt,
is determined
and ~ from
Xt,
(basis)
probability y one half Xt+l
bility
walk
walk
code paths
fractional
the jlow
method
and Sinclair’s
is derived
by means
[2, 4].
control
any specific
The technique
congestion
of certain
turn
of these paths
4). This gives an alternative
condition
differential
as follows: ●
path
existence
of the expansion
through
(Section
in
arguments
the length
of cut set expansion.
enforcement
can be used to define
any pair of bases on the bases-
such that
congestion
can be bounded
known The algorithmic
graph,
[6, 14, 19],
congestion
[9, 16], and coupling
paths between
exchange
to bound
of combinatorial
of the state-space
path
random
show that balance
walk, Broder’s yielded,
convergence
rate
cou-
roughly, [4]. For
regular
matroids,
ments yielded
Dyer
and Frieze’s
O ((n log m+log
geometric
e–l )m4n4)
argu-
●
probability
We analyze
a modification
rate
(Theorem
5.2).
for matroids
an exponentially ●
of the natural
and show O ((n log m + log e-l )n3)
bound
For regular
This that
bound
of this
matroids
an alternative
are tedious
e-l)mn4)
sampling
walk
but
paper),
(details
straightforwhich
running
proportional
theorem
[28] (involving
terminants
and arithmetic
modulo
ning time
O (mn4 log m).
However,
scheme is worse by a factor
time.
There
is
evaluation primes)
and much easier to implement time
braic nature;
graphic
implement
matroids
we need
satisfy
or fail to satisfy
torial
each step of the modified
along the lines of Fredrickson O((n
random
O(log
pling
n)
scheme
.E(C) time
graph.
O(nm)
sented
ploits
●
For
the
cover
C’ is the
scheme
However,
when in
we obtain
is the
counted (More
number only
generally,
ments
for
crease
the
convergence
but
walk
unchanged.)
O (m’n4
The
the
a balanced
walk,
Kirkhoff’s
this
leaves tree
log m),
ability
edges without
matrix
graph
with
when
follows
of the
further
a large
introduction matroid
can
Achieving
2
rates.
scheme
troids
with
edges
family
correlation
in the family
are 5.3.
generally,
parallel
ele-
lated,
significantly
in-
are balanced
all regular
matroids
and the fact that
implies
that
all reguler
their
a large
as “cycles”
less efficient,
number
matroids
for
sat-
all ma-
For exam-
graphic cannot
of parallel
imal
time makes
28
are negatively minors
are balanced.
proof
We
of the fact that
and cocircuits
and “cut s“, by analogy
matroids.
corre-
are also regulm
are balanced.
We refer to the circuits
on
to implement.
any increase in the running
that
Then
as well.
matroids
give here a new combinatorial
based
of matroids
property.
regular
is both
directions
Balance
random
theorem
then tighter
ple, graphic matroids whose bases are the edge sets of spanning trees in a given graph are balanced. More
where
Theorem
scheme
a decom-
7 summarizes
and outlines
random
alternative
Section
5 to
schemes
cases from
2 and implies
natural
of the
Section
in certain
for the modified
rate
in
6 introduces
a minor-closed
isfy the negative
number
time,
of
used
research.
Consider
parallel
of
in the bases-exchange
follows
work
and
the ratios
Section
in Section
of this
from
rates of two sampling
that
3 ex-
condition
4 uses the existence
paths
on convergence
the context
expecta-
a large
from
and more complicated
to introduce
bounds
and we
Section
one pre-
the
) running
the bounds The
the
a sampling
of edges
once;
the construction
matroids,
derived
are then
matroids.
property
that
We give a combina-
a cert ain ratios
mat things
paths
as follows.
classes of matroids
to balance.
the convergence
position
up analogously.
O ((n log m’ + log e-l )@n3 m’
time
proofs
for regular
to construct
for balanced
[4] with
we introduce
case of graphs
edges
cover than
the
blows
in the
sam-
Broder
were of alge-
Section
expansion.
these
bound
thus runs in expected
faster
edges time
latter
of parallel
where
alternative
can be used
Previous
balance:
to satisfy
fractional
condition
can be implemented
is m
that
for regu-
rates for cer-
with
of balance
balance
graph;
walk, For
of balance
the convergence
counter-examples
establish
to
results
time.
[2] and
is clearly
of parallel
of the
There
That
and
here.
number
each step
to Aldous
time,
which
running
[11].
due
running
underlying
tion
walk,
time
time
random
[11],
log rn + log c-l)@n3)
natural in
proof
discuss
necessarily
O(~)
proof
of the paper is organized
run-
certain
For
con-
e.g. see [5] for the special case of graphic
2 is concerned
OIU
so
probability
matroids.
tight). ●
matroids.
The remainder
(furthermore
is not
regular
Section
even though
present),
with
a structure
reduce
of de-
with
by
edges proportional
of its presence in a random
exhibiting
to significantly
of n, it is very simple con-
of the running
of parallel tree is output
to that
failure
of the graph.
lar matroids, tain
results
tree in
can be represented
of the edge remaining
each spanning
figuration
scheme based on Kirkhoff’s
tree matrix
analysis
even for
time to imple-
random
for the full
a number
We give a combinatorial
we need O (inn)
implementation
in O((n log m +log
our
known
polynomial
that
a spanning
each edge has an associated
to the probability
large ground-set.
and are left
ceptually
random
to o@Ut
(such probabilities
introducing
convergence
is the first
remains
ment each step of the modified ward
for example,
where
a graph
convergence
rate [8].
walk
it possible,
of a matroid
with
A cycle is thus a minimal
be augmented
to a basis, while
set whose complement
the case of set which
a cut is a min-
does not cent ain a basis.
The regular binary
matroids
[28].
It is possible
cycle
C
and
g E C and
are known
defined
matmids,
to assign
each
C(g)
to be the orientable
by the following (1) values
element
g, so that
= O otherwise;
and
for each cut 11 and each element if g E D and
D(g)
C(g)
for each
C(g)
=
(2)
most
&l
if
D(g)
D(g)
= +1
and cuts D containing
With ~c(9)D(9) 9 for
all cycles
of graphic to
all
C and
edges
each
in
cycle
setting
on whether direction;
graph,
of the
two
edges the
of each
graph
edges
in
the
D
its
cut
on whether
to
~,
setting
~,
from
the sum over all
A
is zero.
by removing
can be obtained
one element
basis IV defines complement
cut
DN
We refer
of IV.
from
as near-bases.
a unique
Every
unicycle
U defines
is the following.
D~(e)D~(~)
bases
Every
near-
cent ained
a unique relating
in the can be
one element
in U. A useful property
and unicycles
from
to sets which
bases by adding
1
ll~(f)
=
Cu(f)
as unicycle Cu
near-bases
If U = iV U {e, f},
then
Intuitively, unicycles
tend
in opposite containing tions,
that
are zero, while # O, then
is a basis
and
tential result
The
equality
fact
that
or equivalently,
of the
if one of them
Aef
=
non-zero
terms
matroids,
coincides
whether
a quantity If a unit
as a potential
from
the
the the-
resistance
is asof IBI
of e, then the po-
before
of j is Aef.
The
for graphs
with
drop in [5]. We use indices
subsets of B satisfying
concerning
direc-
and a current
the endpoints
has been shown
defined
cuts
tend to be tra-
we can show that
with
networks:
drop between below
the presence
certain
con-
or absence of certain
in the chosen bases. 2.1
The buses of a regular
= lBellBfl
Proof.
is non-zero,
is a basis,
an important
two
same or
could be
+ Cu(~)DN(~)
# O, giving
= ~ llN(e)DN(~) N
:from
from near-bases
on B to indicate
rnatroid
satisfy
-A~f
DN(~)/ZIN(e)
From
quantity,
=
B’\{e}
If e # ~,
in the
sums
follows arise
from only
(B, B’)
we let
U {g},
B“
=
~ l?= x I?.f,
B U {e}\{
and assign a weight
equal to C~uie}(g)Dw\ie](g). B“ zero, then the resulting in Be and 13zt respectively,
= - ~ Cu(e)Cu(~). u expressions
pairs
we ob-
tain pairs (1?”, l?’”) ~ 23. x Z3zf, by means of an exchange involving e and an element g + e. More
so U\{t}
we let A,f
Aef
e, ~ arising
directions,
specifically, define
the quantity ~ in the
enters and leaves at the endpoints
-CU(e)/C~(~). We can now
b(f).
then if all four quantities
iV U {e}
C’u(~)
e and
signed to each edge in the graph,
lBl”lBe~l
say D~(e)
~ U:eGCU
matroids,
to traverse
ory of electrical
Theorem
involved
=
e, f arising
Aef
sum property
follows
~N(~)
cycles cent aining
For graphic
quantity
both in DN and in Cu are e and ~, so that by the zero O; the above equality
the above
versed by e and ~ in the same or in opposite
elements
we have Cu(e)DN(e)
~
for graphic
whether
ditions
the only elements
=
measures
= -Cu(e)Cu(f).
To see this, note that
then
N:eEDN
so
~ to A, it follows
We refer to sets which
to the cycles and
IV U {e, f},
=
a set A
Since a cycle
that
the value of Aef.
simply
C(gp(g)
A.f
the edge is tra-
direction.
=
of
the conditions
affecting
if e belongs U
can easily
g#e
of times
g of C(g)D(g)
contained
separate
A
and
become
conventional
complement
from
~
traverse
a cut the same number
to ~ as from
cycles.
cut
from
= ~ 1 depending
obtained
then
in the
violating
and without
condition,
condition
case
depending
edge is traversed
This
the signs of all elements
direction directions,
in
traverse
in the
underlying
versed in the conventional will
Intuitively, a conventional possible
the
all
-D.
assign
one
this
equations
= +1 for each edge traversed the
of vertices traverse
the
C’ in
C(g)
D(g)
cuts
matroids,
D(g),
cuts involved
= o
it.
by changing
chosen cycles or cuts, without on C(g),
importantly,
For the theorem
U = N U {e, f}.
below, it will be convenient to select a specific element e and require that D(e) = –C(e) = 1 for all cycles C be enforced
values
g, so that
= O otherwise;
pairs iV, U such that
property
cannot The
the with
to this
above
weight e}(9).
=
exchange
If this weight is nonand B’” are indeed bases and a non-zero Wei@
occur here for g = ~ since DB~\fel(.f)
%’\{e}(9)cB’’’u{
29
and B“~
g},
can From
also
be
= 0.
expressed
the point
of view
as of
a pair
(B”,
zero for
l?’”)
some g #
B = B“\{e} a pair
6 Be x &f, e, ~, then
weight
the reverse
and 1?’ = B’” U {e}\{g}
U {g}
(B, B’)
if this
is non-
3
From
exchange
Balance
pansion,
gives back
to
and
Ratios,
Fractional
Match-
ings
6 2.%x Z3~f. Therefore
In this section troids.
we derive
The proof
expansion
is inductive
for balanced
on the rank
intermediate e step, relies on a certain
pei. p=fl =
ratios
E
(z
xf%f9#e
(B’’,l?’’’)et,% =
(x
~
z
~
I&l
(~ CBu{e}(g)DB’\{e} g#e
~131/\{e}(f))(
“ l~ejl
+
~
follows.
theorem
troids
matroids SS that
D implies
correlation
troids
not containing
Some additional
over GF[2] ). There that
Ss as a minor
matroids
property
that
have
to constitute
is a binary ma-
are balanced
[22].
violate
the negative
been
found
counter-examples of the graphic
spanning subgraphs as well as for transversals.
if we consider
edges,
the
path,
5 with add and
connected pr[ef]
the
each an
> ~”
of the dual of
subgraphs ~
and a transversal
consisting
replaced
e joining
a self-loop
vertex
neighbor-
graph
G(M)
if for every element
~atios
e is
e c S the
[18] :
f
by the
with
For
every
balanced
graph
G(M)
matroid enforces
M(S,
B),
ratios.
Let AC t?= and let m~ = VBCA AeieB,eixe
Proof. Note that
the set of bases in Be satisfying
e;.
mA is pre-
mA
is precisely
condition
the set I’e(A).
is equivalent
Hence the first
ratios
to
Pr[mAIZ] Analogously,
> l?r[mAle]
for A ~%,
and note that
(1)
.
= VB6A Aei~B,ei #e ~>
let w
the set of bases in B= satisfying
~
is
the set A, and the set of bases in 23, satisfying
~
is
condition
is
the set I’e(A). equivalent
Hence the second ratios
to
in-
of a path two
paral-
endpoints
anywhere,
then
of
In turn, below:
the last two conditions
follow
from the lemma
for
6 edges we have
The matroid just of the dual of the graphic
to be a truncation matroid
For
of fixed
= Pr[e] Pr[f].
as a truncation
can also be shown
graph
edge
edge
add
spanning
= ~
described
(forests
(connected [26],
[24]
t o nega-
y) and for truncations
lel
3.1
element
corre-
all binary
recently
for truncations
ma-
the negative
of fixed cardinalit
of length
S
of the binary
the graphic cardinality) stamce,
Lemma
I’e denote
the
corre-
cisely the set A, while the set of bases in B? satisfying
[23]; it is known
tive correlation
Pr[e~]
and hence balance
does not satisfy
property
and shown
holds
denote
a specific
The bases-exchange
the bases exchange
are a subclass
lation
correlation
following
Z%, E)
where edges that
not involving
Let fiuther
bases-exchange
(9))
matroids.
(i.e., vectorial
matroid
= G(Be,
of G(M)
of
sp ecif-
cB’’’U{e}f))))
immediately
for regular
Regular
let Ge(M)
subgraph
with
B)
More
‘~f
Pr[e] Pr[ f ], so negative
follow
G(M),
enforcement
expansion.
M(S,
ma-
and, in an
B/ff~~
and the theorem The
graph
said to enforces
B’tEt3e =
a matroid
are omitted.
=
+(
for
hood in Ge(M).
X f3~
z (B,Bf)@FX13ef
ically,
to bipartite
spond to exchanges ‘B’’\{e}(9)cB’’’u{e} (9))
DB1/\{e}(f)cBw”{e}(f)
(B’’,B’’’)@3e
analogous
bipartite
(B’’,B’’’)623ex17zt i7#e,f +
(9))
‘13’’\{e}(9)cB’’’U{e}
Ex-
Lemma matroid variables
3.2 (Main M{
S, i?),
FOT every
Lemma) any monotone
in S \ {e}
is negatively
property correlated
of the graphic Pr[me]
as well.
30
< Pr[m]
Pr[e]
.
balanced
m over with
the e:
Proof.
We show equivalently
The
reasoning
set.
The case where
(and ity
this
is inductive M
has rank
is also the only
of m is used). =
Pr[~le]
and
Pr[m]
=
Pr[~]
Note
further
that
(i)
that
M
erty
m
Pr[mle],
which
Pr[ml~e].
This
and
The proof
show that
if ml
Lemma
over
disjoint
then
Pr[mlm2]
sets
of variables
< Pr[ml]
that
fractional
bound
A
nonnegative each vertex
graph
if there
weights
to the
in u c U, the
on u is IVI,
sum of the
weights
Corollary
3.3
the
G(U,
to
exists
every
= G(13e, Z?z, E)
Proof. from making
Consider G.
by making IBe I copies
edges between in Ge.
eve~
Enforcement
B),
expansion
any Thus
edge
the
was obtained
and be
walk
to ~;
the
on
the
hence
the by
in
the
number
of
there-
1~1). O that from
?natroid
M,
exchange
variation
the gr’aph
conductance
time
byratios
in [18].
bases
total
c
of
and and
directly
The
each
cutset
proof
balanced
mixing:
bounded
A
. [13\/2;
previously
any
at
C B, there
in
inductive
enforcement
is
A
expected
IC’(A)I
and shows expansion
random
expected
of expectations
passes paths
is rapidly
for
of
paths
edge
1~1/113/ 2 m.in(lA/,
For
means
the
from
some
alternative
3.5
that specific
constructed
z 21A/.
by
random
is @ >
distance
t
=
d(t)
Q(nlog
m
+
log e-l )m2n2.
admits
edges
in E such
a
Proof.
The
bounded
of
that
sum of the weights
balanced
element
the
for
with
in v E V, the
from
the I*I
a fractional
bipartite
graph
copies
of each basis
of each
for
graph
4
matching. G*
is the the
by 1131(1 – @2/2)t
product
in [25], where
transition
Hence,
of the natural
by by l/2mn.
has been
case of symmetric
of the
cut set expansion.
d(t)
NIarkov
probabilities
from random
Theorem walk,
3.4 @ can
D
From
Fractional
Path
Congestion
Matchings
to
obtained
basis
of adjacent
G(M)
above
@ in the
distance
B),
in f3=,
in Z3z, and including
of each pair
of ratios
M(S,
variation
and the definition
on v is IUI. matroid
total
conductance
chains
of edges
e E S, the bipartite
admits
all copies
such
through 4.2).
that,
define
of bases
congestion,
An
be bounded any
and for
M(S,
has cutset
can
A is at most
Ic(A)I
can
an assignment
of edges incident
G.(M)
of M
by linearity
on path
l/2mn,
ground-set,
V, E)
and for each vertex
For
to it,
are satisfied.
matroid
4 we argue
through
but,
leaving
natural
properties
pair
. 1~1 paths
Corollary
3.1.
can be extended
from
equal
Pr[m2].
a bipartite matching
incident
of the lemma
and 7n2 are two monotone
of each
correspond
matching
balanced
, we
(Corollary
C(A);
paths
that
Section
leaves
G(M)
00 Remark:
Say
> Pr[~le]
every
paths
lB1/2
are
Pr[nz[~e],
+Pr[~]
We argue
Dalancea ‘ Matroids Tovnds
IBM
Almaden
650 Harry
Milena
FedeF
Center
Research
Road,
Bell
San Jose,
CA
445 South
95120
Abstract
We establish strong expansion properties for the basesexchange graph of balanced matroids; consequently, the set of bases of a balanced matroid can be sampled and approximately counted using rapidly mixing Markov chains. Specific classes for which balance is known to hold include graphic
and regular
matroids.
Morristown,
the set of vectors.
A subclass
lar
will be of interest
Introduction
and
they
matroids:
graphic forth
matroids
denoted
nected
whose vertex-set
S be a finite
ground-set,
tion of subsets of S. Following the set B is said to form matmid
&f(S,
is the collection
operation
the same cardinality
element:
The bases-exchange
has been studied
graph
combinatorial
contexts
related
I?2 = 131\{e ~~}.
properties
for any pair
exchange
property
is that
terminology,
e E B indicating
of bases of a
matroid
of bases J31 and holds:
an ~C l?2 such that ample
standard
for
all
131\{ e}U{~}
of graphic
of G(M).
ple is that maximum
from
the event
e is in the chosen basis. the negative
The cor-
if the inequality
linearly
ex-
and whose bases Another
examis
and whose bases are
independent
< Pr[e] Pr[~]
exists
whose ground-set
matmids,
Pr[e~]
whose ground-set
matmids,
over some field
cardinality
there
is in t3. A main
trees of the graph,
of vectorial
a set of vectors
at random
of S, let e denote
&f (S, l?) is said to satisfy property
relation
l?2 the following
e E l?l
is the set of edges of a given graph are the spanning
that
before in
[12, 20]; here we focus
the mnk of the matroid),
(called
and (ii)
of
of removing
and adding one ground-set
23, and e is an element
if (i) all sets in B have
hence-
by an edge if and only if 132 can be obtained
and let B be a collec-
the collection
23) if and only
M,
by Edmonds
two bases B1 and 132 con-
If B is a basis chosen uniformly Let
All
[28].
131 by the fundamental
on expansion
Preliminaries
of regu-
over every field.
was introduced
with
matroids
is that
of a matroid
gmph
by G(A4),
[10] as the graph
of vectorial
in this paper
are regular
bases of the matroid, from
NJ 07960
are vectorial
The bases-exchange
various 1
Research
Street,
which We introduce the notion of ‘balance” , and say that a rnatroid is balanced if the matroid and all its minors satisfy the property that, for a randomly chosen baais, the presence of an element can only make any other element less likely.
Mihail
Communications
subsets of
holds for all pairs of distinct
elements
ize that negative
is equivalent
Pr[e],
correlation
thus expressing
ence of an element less likely;
the intuitive
studied
before
graphic
matroids
in
fact
~ can ordy make
the negative
correlation
[5, 23].
to Pr[el ~] < that
the pres-
another
element
property
Regular
are known
e, f in S. Real-
and
e
has been
in particular
to be negatively
corre-
lated. *Work
done
primarily
while
the
author
was
at Bellcore.
Here
Permission to copy without fee all or part of this materisl is granted provided that tha copias are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appaar, and notica is given that copying is by permission of the Association for Computing Machinery. To copy otharwisa, or to rapublish, requiras a fee and/or specific permission. 24th ANNUAL ACM STOC - 5/92/VICTORIA, B. C., CANADA @1992 ACM ()-89791 -51 2-719210004/0026 . ..$1.50
we strengthen
lation
to that
is 6alanced
M(S,
B)
itself,
satisfy
minor
the
then
or those this
selecting
operation
if
all
negative
notion
of negative
We its
say that
minors,
correlation
is obtained
operation
that
of selected
26
the
of a matroid
forming and
the
of “balance”:
either
do not.
those
including
M
property.
(A
repeatedly
an element
bases
that
In the case of graphic
corresponds
edges from
by
of choosing
the
to contraction graph.)
corre-
a matroid
pere E S
cent ain matroids, or deletion
e
We first exchange
establish graph
(Theorem
G(M)
both
of G(A4 ), the number
partition
classes is at least
as large
partition
sion properties
had been conjectured
class.
of the to
as the size
Such strong
expan-
by Mihail
[19], in fact for signific=tly
graphs.
M
of edges incident
of the smaller Vazirani
the basesmatroid
1, i.e., for any bipartition
has cutset expansion vertices
3.4) that
of any balanced
wider
correlation
property
negative
correlation
~ implies
and an arbitrary
monotone
between property
and, in view
of the fact
anced matroids, gorithm
e and
an element
e
m on S\ {e},
●
for
their
of self-reducibility an efficient
to approximately
Our results
troid
schemes
we obtain
of any balanced
and
classes of
for two elements
approximation
count
matroid.
of bases of a matroid
The core of our proof here is to show that the
negative
Monte-Carlo
the number
In general,
can be efficiently
for bal-
randomized
al-
of bases
exact
is #P-complete
thus show that
size,
counting
[27].
the set of bases of a ma-
sampled
and approximately
counted
if balance
can be established
for the matroid.
Balance
is known
to hold for graphic
and more gener-
ally regular
matroids;
some counter-examples
to bal-
ance are also known. Pr[em] which
implies
hood
an enforcement
analogous
subgraphs ment
to
(formalized
(this
Previous
encode paths the discrete geometry
neighborfor
as a ‘(ratios finally
implies
for expansion
using
elements
and bound version
certain enfome-
last step was observed
arguments involved
We further
expansion
in [18]), which
condition”
,
of vertex
bipartite
on G(M)
sion for G(M) graphs
< Pr[e] Pr[m]
expanin [18]).
of isoperimetries
from
and the path proof
to
bases-exchange known
significance graph
derives from
ideas, see for example
of the bases-exchange
of the
a sequence of well
[1, 3, 9,14,15,
19, 25],
Consider
the natural
on G(M):
with
random
is the state
If Xt
one half Xt+l
e from if X’
Xt = Xt\
wise Xt+l chain Xt
{e} u {f} = Xt.
uniform import
oracle),
proaching
●
In turn,
reducible
efficient
say, an Most
of G(A4 ) suggests that Xt
and hence possesses the rapid amounts,
roughly,
distribution
[81 ). Therefore,
uniform sampling balanced matroid.
(given,
to the
which
can
it
other-
the Markov
converges
its stationary
on G(M)
efficiently
and that
has large conductance,
for t= poly-log(
= X’,
and,
be used
to Xt
arbitrarily
the natural as an efficient
apclose
random almost
scheme for the set of bases of any
here are of different
Diaconis can yield
combinatorial
uniform populations
sampling yields
that
walk
natural
random
●
c is
The
cardinality
a bound
on total
a technique
O ((n log rn+log
than
(Theorem
5.1).
matroids, of
O((n log rn+log
version
of the
significance
where
the
variation
of
n is the
ground-set,
distance,
In particular,
and conduct~ce
and
we obrandom
E–l )mn2 ) convergence this
arguments
C-* )m2n4)
result
applies
rate to
and significantly imbounds. For graphic
for the nat Ural random
efficient
conductance
rate for the naturzil
algorithmic
graphic and regular matroids proves upon previously known pling
of
[17] we show that
rather
scheme based on the natural
walk with
for self-
symmetry
is as follows: balanced
tain a sampling
to en-
to possess symmetry
convergence
arbitrary the
was Jerrum
type; in fact, matroid
[7] and Mihail
walk.
previously
strong
by extending
m
is
with
as well as for a modified
improvement
For
rank,
only
have been successful
appear
uses paths
a sharper
random this
do not
and Strook
these
any edge
graphs [6, 14, 19]. Our path
Furthermore,
an analysis
matroids, ahnost
graphs
the
and
of using the state-space
arguments
proba-
the ratios
through
The
[14]; such arguments
graphs
from
path congestion
argument
e.g. matching
properties.
Choose
to bound
used here
matchings,
properties, related
over 23 (it is symmetric).
antly, the expansion property
Xt+l
...
edge
of establishing
of paths
graph.
only for combinatorial
t, then
and with
It is easy to see that
distribution
mizing
at time
at random
B then
c
t = 0,1,
as follows:
S uniformly
can be simulated
independence
= Xt,
is determined
and ~ from
Xt,
(basis)
probability y one half Xt+l
bility
walk
walk
code paths
fractional
the jlow
method
and Sinclair’s
is derived
by means
[2, 4].
control
any specific
The technique
congestion
of certain
turn
of these paths
4). This gives an alternative
condition
differential
as follows: ●
path
existence
of the expansion
through
(Section
in
arguments
the length
of cut set expansion.
enforcement
can be used to define
any pair of bases on the bases-
such that
congestion
can be bounded
known The algorithmic
graph,
[6, 14, 19],
congestion
[9, 16], and coupling
paths between
exchange
to bound
of combinatorial
of the state-space
path
random
show that balance
walk, Broder’s yielded,
convergence
rate
cou-
roughly, [4]. For
regular
matroids,
ments yielded
Dyer
and Frieze’s
O ((n log m+log
geometric
e–l )m4n4)
argu-
●
probability
We analyze
a modification
rate
(Theorem
5.2).
for matroids
an exponentially ●
of the natural
and show O ((n log m + log e-l )n3)
bound
For regular
This that
bound
of this
matroids
an alternative
are tedious
e-l)mn4)
sampling
walk
but
paper),
(details
straightforwhich
running
proportional
theorem
[28] (involving
terminants
and arithmetic
modulo
ning time
O (mn4 log m).
However,
scheme is worse by a factor
time.
There
is
evaluation primes)
and much easier to implement time
braic nature;
graphic
implement
matroids
we need
satisfy
or fail to satisfy
torial
each step of the modified
along the lines of Fredrickson O((n
random
O(log
pling
n)
scheme
.E(C) time
graph.
O(nm)
sented
ploits
●
For
the
cover
C’ is the
scheme
However,
when in
we obtain
is the
counted (More
number only
generally,
ments
for
crease
the
convergence
but
walk
unchanged.)
O (m’n4
The
the
a balanced
walk,
Kirkhoff’s
this
leaves tree
log m),
ability
edges without
matrix
graph
with
when
follows
of the
further
a large
introduction matroid
can
Achieving
2
rates.
scheme
troids
with
edges
family
correlation
in the family
are 5.3.
generally,
parallel
ele-
lated,
significantly
in-
are balanced
all regular
matroids
and the fact that
implies
that
all reguler
their
a large
as “cycles”
less efficient,
number
matroids
for
sat-
all ma-
For exam-
graphic cannot
of parallel
imal
time makes
28
are negatively minors
are balanced.
proof
We
of the fact that
and cocircuits
and “cut s“, by analogy
matroids.
corre-
are also regulm
are balanced.
We refer to the circuits
on
to implement.
any increase in the running
that
Then
as well.
matroids
give here a new combinatorial
based
of matroids
property.
regular
is both
directions
Balance
random
theorem
then tighter
ple, graphic matroids whose bases are the edge sets of spanning trees in a given graph are balanced. More
where
Theorem
scheme
a decom-
7 summarizes
and outlines
random
alternative
Section
5 to
schemes
cases from
2 and implies
natural
of the
Section
in certain
for the modified
rate
in
6 introduces
a minor-closed
isfy the negative
number
time,
of
used
research.
Consider
parallel
of
in the bases-exchange
follows
work
and
the ratios
Section
in Section
of this
from
rates of two sampling
that
3 ex-
condition
4 uses the existence
paths
on convergence
the context
expecta-
a large
from
and more complicated
to introduce
bounds
and we
Section
one pre-
the
) running
the bounds The
the
a sampling
of edges
once;
the construction
matroids,
derived
are then
matroids.
property
that
We give a combina-
a cert ain ratios
mat things
paths
as follows.
classes of matroids
to balance.
the convergence
position
up analogously.
O ((n log m’ + log e-l )@n3 m’
time
proofs
for regular
to construct
for balanced
[4] with
we introduce
case of graphs
edges
cover than
the
blows
in the
sam-
Broder
were of alge-
Section
expansion.
these
bound
thus runs in expected
faster
edges time
latter
of parallel
where
alternative
can be used
Previous
balance:
to satisfy
fractional
condition
can be implemented
is m
that
for regu-
rates for cer-
with
of balance
balance
graph;
walk, For
of balance
the convergence
counter-examples
establish
to
results
time.
[2] and
is clearly
of parallel
of the
There
That
and
here.
number
each step
to Aldous
time,
which
running
[11].
due
running
underlying
tion
walk,
time
time
random
[11],
log rn + log c-l)@n3)
natural in
proof
discuss
necessarily
O(~)
proof
of the paper is organized
run-
certain
For
con-
e.g. see [5] for the special case of graphic
2 is concerned
OIU
so
probability
matroids.
tight). ●
matroids.
The remainder
(furthermore
is not
regular
Section
even though
present),
with
a structure
reduce
of de-
with
by
edges proportional
of its presence in a random
exhibiting
to significantly
of n, it is very simple con-
of the running
of parallel tree is output
to that
failure
of the graph.
lar matroids, tain
results
tree in
can be represented
of the edge remaining
each spanning
figuration
scheme based on Kirkhoff’s
tree matrix
analysis
even for
time to imple-
random
for the full
a number
We give a combinatorial
we need O (inn)
implementation
in O((n log m +log
our
known
polynomial
that
a spanning
each edge has an associated
to the probability
large ground-set.
and are left
ceptually
random
to o@Ut
(such probabilities
introducing
convergence
is the first
remains
ment each step of the modified ward
for example,
where
a graph
convergence
rate [8].
walk
it possible,
of a matroid
with
A cycle is thus a minimal
be augmented
to a basis, while
set whose complement
the case of set which
a cut is a min-
does not cent ain a basis.
The regular binary
matroids
[28].
It is possible
cycle
C
and
g E C and
are known
defined
matmids,
to assign
each
C(g)
to be the orientable
by the following (1) values
element
g, so that
= O otherwise;
and
for each cut 11 and each element if g E D and
D(g)
C(g)
for each
C(g)
=
(2)
most
&l
if
D(g)
D(g)
= +1
and cuts D containing
With ~c(9)D(9) 9 for
all cycles
of graphic to
all
C and
edges
each
in
cycle
setting
on whether direction;
graph,
of the
two
edges the
of each
graph
edges
in
the
D
its
cut
on whether
to
~,
setting
~,
from
the sum over all
A
is zero.
by removing
can be obtained
one element
basis IV defines complement
cut
DN
We refer
of IV.
from
as near-bases.
a unique
Every
unicycle
U defines
is the following.
D~(e)D~(~)
bases
Every
near-
cent ained
a unique relating
in the can be
one element
in U. A useful property
and unicycles
from
to sets which
bases by adding
1
ll~(f)
=
Cu(f)
as unicycle Cu
near-bases
If U = iV U {e, f},
then
Intuitively, unicycles
tend
in opposite containing tions,
that
are zero, while # O, then
is a basis
and
tential result
The
equality
fact
that
or equivalently,
of the
if one of them
Aef
=
non-zero
terms
matroids,
coincides
whether
a quantity If a unit
as a potential
from
the
the the-
resistance
is asof IBI
of e, then the po-
before
of j is Aef.
The
for graphs
with
drop in [5]. We use indices
subsets of B satisfying
concerning
direc-
and a current
the endpoints
has been shown
defined
cuts
tend to be tra-
we can show that
with
networks:
drop between below
the presence
certain
con-
or absence of certain
in the chosen bases. 2.1
The buses of a regular
= lBellBfl
Proof.
is non-zero,
is a basis,
an important
two
same or
could be
+ Cu(~)DN(~)
# O, giving
= ~ llN(e)DN(~) N
:from
from near-bases
on B to indicate
rnatroid
satisfy
-A~f
DN(~)/ZIN(e)
From
quantity,
=
B’\{e}
If e # ~,
in the
sums
follows arise
from only
(B, B’)
we let
U {g},
B“
=
~ l?= x I?.f,
B U {e}\{
and assign a weight
equal to C~uie}(g)Dw\ie](g). B“ zero, then the resulting in Be and 13zt respectively,
= - ~ Cu(e)Cu(~). u expressions
pairs
we ob-
tain pairs (1?”, l?’”) ~ 23. x Z3zf, by means of an exchange involving e and an element g + e. More
so U\{t}
we let A,f
Aef
e, ~ arising
directions,
specifically, define
the quantity ~ in the
enters and leaves at the endpoints
-CU(e)/C~(~). We can now
b(f).
then if all four quantities
iV U {e}
C’u(~)
e and
signed to each edge in the graph,
lBl”lBe~l
say D~(e)
~ U:eGCU
matroids,
to traverse
ory of electrical
Theorem
involved
=
e, f arising
Aef
sum property
follows
~N(~)
cycles cent aining
For graphic
quantity
both in DN and in Cu are e and ~, so that by the zero O; the above equality
the above
versed by e and ~ in the same or in opposite
elements
we have Cu(e)DN(e)
~
for graphic
whether
ditions
the only elements
=
measures
= -Cu(e)Cu(f).
To see this, note that
then
N:eEDN
so
~ to A, it follows
We refer to sets which
to the cycles and
IV U {e, f},
=
a set A
Since a cycle
that
the value of Aef.
simply
C(gp(g)
A.f
the edge is tra-
direction.
=
of
the conditions
affecting
if e belongs U
can easily
g#e
of times
g of C(g)D(g)
contained
separate
A
and
become
conventional
complement
from
~
traverse
a cut the same number
to ~ as from
cycles.
cut
from
= ~ 1 depending
obtained
then
in the
violating
and without
condition,
condition
case
depending
edge is traversed
This
the signs of all elements
direction directions,
in
traverse
in the
underlying
versed in the conventional will
Intuitively, a conventional possible
the
all
-D.
assign
one
this
equations
= +1 for each edge traversed the
of vertices traverse
the
C’ in
C(g)
D(g)
cuts
matroids,
D(g),
cuts involved
= o
it.
by changing
chosen cycles or cuts, without on C(g),
importantly,
For the theorem
U = N U {e, f}.
below, it will be convenient to select a specific element e and require that D(e) = –C(e) = 1 for all cycles C be enforced
values
g, so that
= O otherwise;
pairs iV, U such that
property
cannot The
the with
to this
above
weight e}(9).
=
exchange
If this weight is nonand B’” are indeed bases and a non-zero Wei@
occur here for g = ~ since DB~\fel(.f)
%’\{e}(9)cB’’’u{
29
and B“~
g},
can From
also
be
= 0.
expressed
the point
of view
as of
a pair
(B”,
zero for
l?’”)
some g #
B = B“\{e} a pair
6 Be x &f, e, ~, then
weight
the reverse
and 1?’ = B’” U {e}\{g}
U {g}
(B, B’)
if this
is non-
3
From
exchange
Balance
pansion,
gives back
to
and
Ratios,
Fractional
Match-
ings
6 2.%x Z3~f. Therefore
In this section troids.
we derive
The proof
expansion
is inductive
for balanced
on the rank
intermediate e step, relies on a certain
pei. p=fl =
ratios
E
(z
xf%f9#e
(B’’,l?’’’)et,% =
(x
~
z
~
I&l
(~ CBu{e}(g)DB’\{e} g#e
~131/\{e}(f))(
“ l~ejl
+
~
follows.
theorem
troids
matroids SS that
D implies
correlation
troids
not containing
Some additional
over GF[2] ). There that
Ss as a minor
matroids
property
that
have
to constitute
is a binary ma-
are balanced
[22].
violate
the negative
been
found
counter-examples of the graphic
spanning subgraphs as well as for transversals.
if we consider
edges,
the
path,
5 with add and
connected pr[ef]
the
each an
> ~”
of the dual of
subgraphs ~
and a transversal
consisting
replaced
e joining
a self-loop
vertex
neighbor-
graph
G(M)
if for every element
~atios
e is
e c S the
[18] :
f
by the
with
For
every
balanced
graph
G(M)
matroid enforces
M(S,
B),
ratios.
Let AC t?= and let m~ = VBCA AeieB,eixe
Proof. Note that
the set of bases in Be satisfying
e;.
mA is pre-
mA
is precisely
condition
the set I’e(A).
is equivalent
Hence the first
ratios
to
Pr[mAIZ] Analogously,
> l?r[mAle]
for A ~%,
and note that
(1)
.
= VB6A Aei~B,ei #e ~>
let w
the set of bases in B= satisfying
~
is
the set A, and the set of bases in 23, satisfying
~
is
condition
is
the set I’e(A). equivalent
Hence the second ratios
to
in-
of a path two
paral-
endpoints
anywhere,
then
of
In turn, below:
the last two conditions
follow
from the lemma
for
6 edges we have
The matroid just of the dual of the graphic
to be a truncation matroid
For
of fixed
= Pr[e] Pr[f].
as a truncation
can also be shown
graph
edge
edge
add
spanning
= ~
described
(forests
(connected [26],
[24]
t o nega-
y) and for truncations
lel
3.1
element
corre-
all binary
recently
for truncations
ma-
the negative
of fixed cardinalit
of length
S
of the binary
the graphic cardinality) stamce,
Lemma
I’e denote
the
corre-
cisely the set A, while the set of bases in B? satisfying
[23]; it is known
tive correlation
Pr[e~]
and hence balance
does not satisfy
property
and shown
holds
denote
a specific
The bases-exchange
the bases exchange
are a subclass
lation
correlation
following
Z%, E)
where edges that
not involving
Let fiuther
bases-exchange
(9))
matroids.
(i.e., vectorial
matroid
= G(Be,
of G(M)
of
sp ecif-
cB’’’U{e}f))))
immediately
for regular
Regular
let Ge(M)
subgraph
with
B)
More
‘~f
Pr[e] Pr[ f ], so negative
follow
G(M),
enforcement
expansion.
M(S,
ma-
and, in an
B/ff~~
and the theorem The
graph
said to enforces
B’tEt3e =
a matroid
are omitted.
=
+(
for
hood in Ge(M).
X f3~
z (B,Bf)@FX13ef
ically,
to bipartite
spond to exchanges ‘B’’\{e}(9)cB’’’u{e} (9))
DB1/\{e}(f)cBw”{e}(f)
(B’’,B’’’)@3e
analogous
bipartite
(B’’,B’’’)623ex17zt i7#e,f +
(9))
‘13’’\{e}(9)cB’’’U{e}
Ex-
Lemma matroid variables
3.2 (Main M{
S, i?),
FOT every
Lemma) any monotone
in S \ {e}
is negatively
property correlated
of the graphic Pr[me]
as well.
30
< Pr[m]
Pr[e]
.
balanced
m over with
the e:
Proof.
We show equivalently
The
reasoning
set.
The case where
(and ity
this
is inductive M
has rank
is also the only
of m is used). =
Pr[~le]
and
Pr[m]
=
Pr[~]
Note
further
that
(i)
that
M
erty
m
Pr[mle],
which
Pr[ml~e].
This
and
The proof
show that
if ml
Lemma
over
disjoint
then
Pr[mlm2]
sets
of variables
< Pr[ml]
that
fractional
bound
A
nonnegative each vertex
graph
if there
weights
to the
in u c U, the
on u is IVI,
sum of the
weights
Corollary
3.3
the
G(U,
to
exists
every
= G(13e, Z?z, E)
Proof. from making
Consider G.
by making IBe I copies
edges between in Ge.
eve~
Enforcement
B),
expansion
any Thus
edge
the
was obtained
and be
walk
to ~;
the
on
the
hence
the by
in
the
number
of
there-
1~1). O that from
?natroid
M,
exchange
variation
the gr’aph
conductance
time
byratios
in [18].
bases
total
c
of
and and
directly
The
each
cutset
proof
balanced
mixing:
bounded
A
. [13\/2;
previously
any
at
C B, there
in
inductive
enforcement
is
A
expected
IC’(A)I
and shows expansion
random
expected
of expectations
passes paths
is rapidly
for
of
paths
edge
1~1/113/ 2 m.in(lA/,
For
means
the
from
some
alternative
3.5
that specific
constructed
z 21A/.
by
random
is @ >
distance
t
=
d(t)
Q(nlog
m
+
log e-l )m2n2.
admits
edges
in E such
a
Proof.
The
bounded
of
that
sum of the weights
balanced
element
the
for
with
in v E V, the
from
the I*I
a fractional
bipartite
graph
copies
of each basis
of each
for
graph
4
matching. G*
is the the
by 1131(1 – @2/2)t
product
in [25], where
transition
Hence,
of the natural
by by l/2mn.
has been
case of symmetric
of the
cut set expansion.
d(t)
NIarkov
probabilities
from random
Theorem walk,
3.4 @ can
D
From
Fractional
Path
Congestion
Matchings
to
obtained
basis
of adjacent
G(M)
above
@ in the
distance
B),
in f3=,
in Z3z, and including
of each pair
of ratios
M(S,
variation
and the definition
on v is IUI. matroid
total
conductance
chains
of edges
e E S, the bipartite
admits
all copies
such
through 4.2).
that,
define
of bases
congestion,
An
be bounded any
and for
M(S,
has cutset
can
A is at most
Ic(A)I
can
an assignment
of edges incident
G.(M)
of M
by linearity
on path
l/2mn,
ground-set,
V, E)
and for each vertex
For
to it,
are satisfied.
matroid
4 we argue
through
but,
leaving
natural
properties
pair
. 1~1 paths
Corollary
3.1.
can be extended
from
equal
Pr[m2].
a bipartite matching
incident
of the lemma
and 7n2 are two monotone
of each
correspond
matching
balanced
, we
(Corollary
C(A);
paths
that
Section
leaves
G(M)
00 Remark:
Say
> Pr[~le]
every
paths
lB1/2
are
Pr[nz[~e],
+Pr[~]
We argue
