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
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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[ml~]
pothesis
on
237 for
variable
would
(iv)
by
Pr[ml~e]
Pr[~]
the
such that
that
(iv)
Pr[rnl~e]
< Pr[m[f]
applying
to
the
the
1,
Pr[ml~e]
Pr[m\~]
there
holds.
by
= Pr[m]
exists
always
and
with
note
3.4
e]
In
between
any
number
of
o), which
is equivalent
is finally
equivalent
completes
the
to Pr[nz[fe]
to
proof
Pr[ml~e]
z
of Lemma
<
A:
and (iii).
element
IAI
path
~
3.2,
(with
Pr[fle]
>
> 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
<
f#e
for some ~, Pr[flme]
of P that
graph
matchings
Remark:
Hence
For
Proof.
fore
f#e
of edges
for a fractional
fractional
most
lemma
(iv),
by (ii)
some
In particular,
hy-
Pr[m[
(i)
the copies
1.
it was the
principles:
By identifying
to each edge of E a weight
the bases-exchange
(iii)
m
the
P.
G* has a
prop-
inductive
then
and hence
c1
fact
and
property
number
the conditions
by ap-
on B~ for
condition,
assigning
Theorem
the
Hall’s
matching
and
to the
that
from
forced
averaging
perfect
Pr[m\~].
Pr[~]
> Pr[nz[~e],
+Pr[~]
We argue
<
G* satisfies
basis
Pr[7nl~e],
to O. If, in addition,
+ Pr[~]
Pr[ml~]
+ Pr~]
f
that
the monotonicstep note
monotone
Pr[ml~e]
follow
Pr[~]
by
~ forced
case that
Pr[ml~]
variable
< Pr[m].
+ Pr~\e]
hypothesis
the
Pr[ml~e]
the
inductive
Pr[m\~e]
(ii)
inductive
with
where
Pr[~\e]
is balanced,
the
Pr[rnle]
size of the ground-
n = 1 is easy to verify
point
For the
Pr[mle]
plying
that
on the
implies
In order
to obtain
are tighter
bases
the
fractional
now
the
previous
31
than
bounds
those
of Corollary
matchings
section
on convergence
to
defined
construct
3.5,
in
rates
we shall
Corollary
paths
that
joining
use
3.3 of
all
pairs
of bases
in the
matroid,
while
anced
the number
keeping
of paths
We thus
wish
exchange
path
bases-exchange
The
dom
permutation
Consider
el,..
a specific
each ~; is either
present
or absent
tination
B’
destination
in B’.
Note
are initially
path
with
i steps,
arranged
with
B’ in the first
B has exactly
number
i elements,
of paths
initially,
when
that
i =
after
are unifornily
O and
i steps,
over
= ’61 ...e~?;+~ and
at f3i that are already
l)th
ing
step.
The
from
23~+1 to
given
steps,
the
paths
arranged
over
t3~ = {B’},
Lemma
thus
with
Bi+l.
ensuring
The
B’
number
The paths
i are those
For
each
tined
with
such
to it
leaving
B’,
before
bases
wit h destination
B’,
B’
required.
The
Note
ements
gives
i,
there
are
in the
consider
a sequence
e2, B2, . . . . em, B~ =
it
agrees
with
uniformly
B
from
in
the
so that
1/2
4.4
The
B’
, B~
chosen
differ
in
ei is present
from
each.
such
a choice
we expect
Since
to
H
expected
from
length
of a path
from
2n.
Lemma
= ~
to be fixed.
~
of two bases con-
of elements
4.3,
is the
. . . , em is the random
were
and
the element
difference
is at most
= B’
B;_l
path
order
where
from
B
in whih
B
to
=
B’
the edges
D
uniformly
Say that
are at
a path
leaving
a
Bel ...ei_l.
Corollary
through
so
Mutliplying
this
desti-
. . . &_l ] as
The
By
[~~1
a basis
all
Now
Lemma
Lemma
number
Bi
in
4,3, where
is chosen
that
E [~~1
Pr~~lel,.
5
From
Path
4.1,
. , &;-l]],
permutations
if this
basis
its destination.
expected
Pr~~l@l,..
over
B and
.,,ei., I paths
of possible
4.5
is not
a basis B = B;l ,,6,,, is at most
Proof.
[BIE
goes through
but
this
the
is chosen
uniformly
from
. . ,ii–1]]
<2n.
Congestion
paths
expectation
the expectation
which
B.
of
2nlt31.
ei’s
is
being
are
chosen.
uniformly
from
Be, ...~i. implies
Cl
to Rapid
Mixing
same
ei
a basis,
Then the expected
.
Bi_ 1 and B differ in e; is at
2n times.
Follows
andel,
1)
Z?el ~i_lz.
...
IB[ paths
des-
IB[ Pr[~lel
of paths
el, . . . . ei_l.
that
n elements,
at most
BO, B1,...
O
that
than
same number
Proof.
in
1231/lBel
Ili?&,.+ej_lzl
quantity
B’
B,
ei is chosen
probability
Corollary
. . . 13~_1].
over
in expectation.
the
each
any B to any
are
all paths
the
arranged
by the number
nations
step
are uniformly
at each of these
quantity
at
(i+
with
@ has only
~ in B&l ...~i at step
some
length
through
so that
If ei is chosen
happen
match...~i.
of paths
a basis
destination
paths
2n.
t ains the
Mel
basis B in t3&, ...&i at step i is [13\ Pr[~[41
of
1t31/z.
of paths
Bo, el, Bl,
is chosen
other
it, then
exchange
are
number
expected
a basis
in B, since the symmetric
B’
where
after
Bi
is on the path
ezpected
i is at most
lerm-na.
Given
of ei such
Proof.
as desired.
4.1
Proof.
most
holds
weights
m steps,
step
of
i
matching
that
destination
After
the
are currently
to the associated
matching,
where
number
induc-
to the minor
4.3
el, . . . . ei and
The paths
there for the
a fractional
the
the following
elements
destination
to each possible
chosen proportionally
fractional
Such
number
. . .&I]).
is at most
number
of bases and elements
after
of a fractional
3.3 applied
the probabilities
Assume
with
at Bi that
Bi+I.
by Corollary
B.
~~+1 = B&l ...ei~.
at 13i+1 are left
paths
at
expected
B’)
bound
B,
agree
basis
condition
to
expected
one
sense that
such
the
The
edge (B,
order
and
Lemma
after
Z?i = 1$+1 u t?&l,
are sent to 13i+l by means
exists
that
This
L?i =
expected
an exchange
the
are uniformly
in the
the paths
arranged
‘;+1
(i+
B’
any
we give
des-
over
ensure
at each
steps is the same for all of them.
B~+l
with
each basis
destination
the
ei is
arranged
paths
with
In
of S.
over Bi = B&l ...ei. the set of bases that
expected
tively
m elements
the paths
B ‘. We shall
that
. . . &~_~],Pr[~le~
4.2
through
a ran-
on whether
destination
the
choice
uniformly
set of all bases B, in that
such
Corollary
one
= B&, ...&~. where
that
using
[131rnin(Pr[e;l&~
M ( S, B),
depending
and
i,
in the bases-
the
B’
step
paths
and
B and a destination
with
., em of the
ei or Z,
lengths
matroid
begins
during
of a bal-
small.
]13[2 paths
of an origin
construction
path
each basis
of a balanced
for each choice
B’.
the
through
to construct
graph
graph
bound
holds
a particular
and
ej
with
for
paths
exchange
i
<
j
entering
involving
can
be
two
used
~.
el-
Consider
exchange
only
32
the
natural
graph
G(M)
random
of
walk
a
on
balanced
the
basesmatroid
M(S,
23), as defined
tion
we
yield
show
bounds
on
ral random
icantly
better
and
those
of Corollary
consider
half
rate
Yt+l
(ii)
is a near-basis
choose
= YtU{~}.
tionary,
the
exactly
1/2.
sampling
Hence
We proceed
between
flow
pij
In
let
Let
random
pair
A and
B of states
paths
use edges
denote
paths
an upper
path
from
been
defined
B.
the
and suppose
that
Let
rAmB
Next
the
a basis
is
distance
at time
an equation
chain
results
along
the
stating
in
Markov
among
hi(t)= Pr[Zt
that
hi’s
at
0:
each step of the M.arkov
averaging
of
of (j, i)’s
such
should
bound
the
discrepancies
that
pji # O; thus
decrease:
pij
can be obtained
(j, i)’s
with
endpoints
we give
= i] –Ti,
and
above
length
state
of the
that
have
one path
that
assigns
A to state
(i, j).
that
is more
significantly
different
of j and
an outline
we shall
the averaging
is more
convergence
of the
effective
rapid)
if it
(and
involves
discrepancies
at the
i:
Now
recall
pair
A and
B of states
these
paths
use edges
PAB
denote
event
that
the
Then
event
A to B.
for the variation
bound
that
edges
t
A to
t:
of the
the
(so that
# O), let
choose
edge
the
stating
along
any
on the probability
a specific
a bound
hence
that
the
PAB
that
Let
B,
paths
and
all paths
is left
as follows.
Let
use hi as short
for
33
1 denotes
one path
assigns
A to
state
edge
that
path
O).
Let
and
the
E PAB
denote
in the path
from
of the path
from
an upper
FinaUy,
defined
bound
recall
that
bet ween pairs
at random
according
mAZB
B.
that
Recall
that
(i, j).
Then
we are using
length
B
probability
on the probability
a specific
maximum
was used
any
(so that
p~j #
A to
and let ij
of lAB.
between
chain
that
from
have been
that
of [7, 21] (except
than
such
the length
that
that
state
bound
cent ains
for the
(i, j)
defined
Markov
path
length
we choose
from
(i, j)
denote
recall
among
of the
was chosen,
edge
expected
have been
chosen
/Ajj
on the
upper
proof
here
#
Assuming
distribution
from
ji’ow
between
all paths
of states
fumlly
discrepancies
for which
chain
that
And
the ergodic
flow.
defined
path
matrix
path
paper;
an
endpoints
the distribution
full
of the
Pji
of
of states
detailed
discrep-
chain
ergodic
ergodic
such
to the
contains
bound
The
such that
i)’s
can be
attributes
y to discrepancy
of (j,
the tot al discrepancy
distribu-
states
on the expected
to the
the following
d(t)
of the
(i, j)
pairs
L be an upper
path
that
analogous
following
of
in terms
any two
been
Finally,
according
chosen
this
bound
between
probability
the
and
Yt is sta
stationary
the
have
A to E.
at random
is
cut
a Markov
measure
denote
w
that
these
an equation
st ationarit
techniques
[17],
to use as a
rate
for
the
= ~ipij
i and j,
let
so
First
from
using
in
choose
space S and transition
w~j is the same between
# O,
ancy
Next,
~.
appeared
the endpoints
D is lD1/2n[13[;
convergence
% denote
wij
states
when
Yt is on
particular,
on state
+ 1),
that
= ~i
that
derived:
Yt+l
is 1/21231, and
cut
that
= Yt \ {e};
Yt is also appropriate
to bound
z~, t=o,l,...
with
the
at random
that
d’(t)
to those
y one
then
Y~+l
Let
modified
~ S is the
of a basis
to check
for t3.
congestion.
of Zt.
Dfi
probability
scheme
P’ = !j(P
and
It is easy to verify
the
The
unifornily
of a near-basis
furthermore
tion
Dy,
probability
probability
path
and
~ from
walk
one half
at random
hard
of
bases
probability
(i) if Yt is a basis
Yt uniforxnly
if ~
random
probability
it is not
modification
even faster.
with
as follows:
First,
natu-
uses both
With
hi(t).
are signif-
3.5 in terms
the
modified
becomes
= Yt, and
Yt, then
Yt+l
this
of the
here
which
Yt is as follows:
determined
e from
walk
sec-
congestion
rate
further
for
In this
path
derived
We
random
walk
on
convergence
the bounds
near-bases;
convergence
bounds
than
natural
random
the
the
walk;
conductance.
of the
in the introduction.
how
here)
L is an
the chosen
along
expected
we get:
to
to the
the
path
lines
rather
d’(t)
– (i’(t
+ 1)
Theorem
5.2
balanced
matmid
bounded
bye
parallel
in
state
the
elements
collapsing;
proportional
of elements
in it,
sum
for
size of the
state
due
l/mti.,
all
we get
the
the
above
and
desired
noting
bound
that
on the
d’(0)
~
variation
distance.
troid
&f(S,
lary
4.4,
B) we have
and
definition
5.1
balanced
matroid
bounded
bye
the
bases
on any balanced
by
Corollary
the natural
POT
the
total
in time
t=
modified
4, except
The
as for
that
the
expectation
4.5.
Paths
whose
by first
random,
tween
bases,
paths
with
thus
due
to paths
the
above
not
hard
to verify
L = (2n
+ 1)/lB[.
J=4n
graph
elements
whose
choosing
increase
as a starting
by
associated
reduces
the
differences.
been
the
factor
To reduce
point
weight,
for
so that
collapsed.
walk,
unlike
of parallel
time.
A basis
random
the
TAD
Therefore
the natural
elements
can
of largest
weight
be found
weight
affect
needed
by
while
for
random
does not
of maximum
walk
elements
and
on any
The
bounds
for
balance
matmid
number
of parallel
maintaining
The
and
greedily
preserving
the modified
remain
random
the same
elements
after
are
added
balance.
algorithmic
significance
5.3 was pointed
path
definition
w =
of
6
Better
out
of Theorems
5.1, 5,2,
in the introduction.
Path
Congestion,
composition,
graph;
De-
Symmetry,
Series
Parallel
are de-
In
uniformly
path
congestion
for the modified
Furthermore,
5.3
arbitrary
by Corollary
basis
are only
the
walk
while
through
near-bases
to
Theorem
in
number
a random
path
endpoints
that
and
in the obvious
paths
2n[Bl
include
endpoints
observations
have
random
the
bases,
in the
may
ba-
of the weights
and n~aX / T~aX .b=i~ is
m is the number
of elements
after
where
the number
selecting
an
graph
of the bases-exchange
reducing
be
between
expected
a neighboring
arbitrary
can
paths
of such
by the
endpoints
then
on any
are used to simulate
is at most
choosing
and
walk
bases-exchange
near-bases
a vertex
latter
at
the
c-l)n’m.
walk,
number
the
fined
and
distance
m+log
random
expected
through
4.2
random
variation
Q(nlog
an edge can be bounded
paths
parallel
to start
ele-
of any given
independence.
the edges of the bases-exchange
way.
m/n,
the running
w = l/[ Z3[nzn, 1= 2n by Corol-
but
number
on the resulting
by n for
weight
increase,
the
transformation
space,
increase
this
view
of elements
This
may
to it.
by m~aX _&&
at most
walk,
ma-
Hence
are defined
Section
walk
L = l/21B[
of L.
Theorem
For
random
a large
avoid
and
a basis of maximum
the modified
For the natural
To
multiplied
we choose
walk
element
to the product
to the
be replaced
can
cause
IS I.
weights
m~.X/rti.
quantity,
may
can be
parallel
the probability
near-bases.
random
using
of the
each
as a weight
sis is now
the
where
on any
e-l)n3.
of elements
elements,
of parallel
cut
case
size
ment
walk
distance
t = Q(n log m+log
elements
space
random
van”ation
number
all parallel
after
total
the
large
These
this
Finally
now
a very
collapse
the modified
the
in time
Consider
have
For
be-
due
to
congestion
bases.
From
of L it
random
l/2nlBl
is
walk
graph
a balanced
matroid,
pair
each
we have
of bases,
near-bases
constructed
and
bounded
Since each basis has n incident
each
element
on each
of the
basis,
edge to be bounded
by
for some a, the n3 factor
random
walk
be-
load
nl 131 on the load
one might
for
paths
the
sis vertices,
for
of about
and
a bound
rate for the modified
34
of bases
edges using
of a IBI holds
+ 2. Thus
consisting
tween
load
and
the
on
of ba-
edges, one
expect
the
1231. If a bound
of convergence
can be reduced
to
an2.
We shall
SUCh
bounds.
the
and
a fractional
with
weight
lZ3e1, and
note
N..
constraint
using
basis
the
in B.,
that
a near-basis
to
for
edge
each
in ~e
N U {f}
to
graph
by the
above
convenience,
if
weflv.
other
For ~
U
O, wef u = O. The
requires
wefN
try
edge
U)e~N = 1~1 for
wefu
by
N
giving
=
then
exact
conservation
have
thus
we now
add these
each
weight
basis,
from
e and
for
resulting
weight
cases,
the
Nit
equation.
with
U with
simple
matching
fractional
constraints,
namely
that
assigns
and
a near-basis.
has the
fractional
exact
weight
[B!.
the
=
sum
weight
the
6.1
the
If a balanced
reduces
matching
sum of m fractional
the
One can also consider
the
which
only
property
always
of the
basis
requires
holds
for
N U {f}
is
Consider
the first
the element
chosen
N U {j}
At
its
The
indeed
mat things,
be
for the first
to N.
matching,
e is chosen
its
path
the n3 factor
random
step,
and
If e = f,
then
the load
and
uniformly
the expect ed load
joining
if the
obtained
the
(i + l)th
the
i elements
then
the load
This
case occurs
fractional
el, e2,.,.
weight
ing
walk
Let
e be
consider
an
the load
of
is determined
it is precisely
at random
Wef N.
from
all
m
is
by
case occurs
load for this
this
35
quantity
if ~ is one of
the first
to N will
contracted
occur
~
IBI in
i/m.
is zero.
Otherwise,
be involved
probability
step is again
over
or deleted,
in N or not.
case.
bounded
aIl m steps
by ~
depend-
This
Since
(m – i)/m,
in the
the chosen
so the expected
this
i steps,
consideration
for some e = e~+l after
they
with
for
probability
m – i elements,
uniform
as the
with
, e; have been
bounded
paths,
e~ used
N U {f}
matching
matroid
each element
of the
for the edge under
on whether
has only
a basis
step
el, ez, ...,
the edge joining
matching
a balanced
one for
k
of the
is a very
these
property
then
and
step of all paths.
edge is ll?~[. If e # f, then
Since
‘ [B[.
there
fractional
edge
pToperty,
(a + 2) IBI
that
edge,
degree
that
and all its minom
by adding,
1231= ]Drvl
to the
matroid
rate foT the modijied
edge joining
this
e c CU.
each
satisfying
to each
can
follows
to cxn2.
Proof.
e E ~N,
matchings.
Note
say that
decomposition
holds,
~ symlnefrom
a-decomposition
is at most
of the convergence
is X= [Z3eI = nlZ31, and
uniform
IB[
We
and
is equal
times
D,N.
matchings
matchings,
fractional
basis or near-basis
e, f, N.
property,
n, since
congestion
conser-
of e. Suppose
is ~eeD~
weight
C
if symmetry
all
decomposition
this
satisfy
e. For
wefu
two
fractional
near-basis,
for each basis
for each near-basis
both
m
of the
i
matchings
the
~
e and
m fractional
each
hold
for
that
Lemma
basis
,U, we set ?UefN =
each choice
each
N with
~ # e
we write
in addition,
constructed
23 to JV, one for
must
wfeN
a-decomposition
elements,
from
~ and
for some a. Note
a=
wef N the
to the
of fractional
= 123e] for given
edge for
we must
e
f
for
property
by the fractional
z
We
this
if wefN
holds,
weaker
IBZ[ to each
matching
of f,~
given
of this
e = j’ is /13jl,
with
= IZ371 for given
with
an element
joining
definition
~ O, and
the weight
weights
N U {f}
graph
mat thing
N U {e, f},
choices
WefN
that
first
to
equations
E
f
z
namely}
basis
associated
denote
fractional
=
matching
the
and
n[t3[.
of N,
the
N
as representing
this bipartite
a fractional
cut
that
add to IB[, and since
Note
is
&
means
edge joining
e
in Me a weight
N in Afe and
the
assigned
vation
e
1231.
e from
this
edges joining
then
bases-exchange
belongs
weight
that
this
have
in Afe does exist.
Given
that
to
[13eI to each basis in Ii?z and weight
near-basis
each
weight
by removing
each near-basis
the
JVe such
of weights,
to a particular
the fractional
can
bases,
each with
assigned
weight
e, and we can infer
weight
cut,
must
joining
e, one
all
at each near-basis
to be satisfied
a corresponding
edges
near-bases
weight
the
If we view
with
the bases in Be can be matched
The
leaving
is simply
e. In terms
can yield
an element
in Afe, simply
113.1, satisfying
of I&l
all
that
of them.
in Be, and
that
between
associated
to the near-bases
each
graph
Given
matching
to their
To see this,
conditions
bipartite
near-bases.
find
belongs
examine
assigned
Consider
bases
now
minor
load
is now
this
second
the expected
]231. Adding
of the paths
(plus
the
.
initial
load
obtain
assignment
a bound
thus
the
ns factor
to bases),
for
the
we
congestion,
convergence
not
The main
obstacle
in obtaining
the
for
fractional
bounds
matchings
plicitly
known.
explicit
construction
We now
is the
that
of f.
If
the
direction
ing
e determines
not
ex-
such
an
6.2
D(e)D(
then
f )Aef
a
>0
for
symmety
regular
well
and
matroids
elements
and
exact
for
hold.
the matmid
components
are
cluded
(as
all
trY
its
K4
or
quantities
are nonnegative
the conservation
wefN
laws:
=
-
~
=
a
DN(e)DN(f)Aef.
These
by assumption,
and satisfy
If e E DN,
matroids
elements
D
characterithat
Fano
matroids,
for
graphs,
as well.
a complete
depending
We do not
that
set wefN
in particular
binary
the
f,
as before
have
matroid,
satisfies
symme-
on whether
of N
form
an
an exboth
a basis
with
{e, f}).
graph.
We
hold
regular
one of the
hold
is not
since
for
in traversing
are series-parallel
(settingWe f N = 4,6
or only
on graphs
either
minor
then
for the minors
above
decomposition,
end-
f,
e, f in travers-
used
They
hold
The
the
with
decomposition
of the balanced
exact
The
and
matmids
zation
e, f,
decomposisition
the gmphic
Proof.
x
D
sat@es
decomposition
that
holds
biconnected
series-parallel
matmid
cuts
exact
are precisely
whose
all
such
as symmetry)
minors
regular
swaps
through
direction
all its minors
properties
but
is an isomor-
an endpoint
graphs.
and since
the two
can be obtained.
If
the
there
e fixed
and exact
Remark:
Theorem
leaves
e shares
for series-parallel
K4,
because
used by a cycle
so symmetry
a case where
vertices,
of K.
fact
are in general
give
any
points
for the val-
a-decompositions
share
phism
rate
O
ues of a achievable
that
near-bases
of (a + 2)1231 on the path
reducing
to cm2.
from
the
anced
then
know
whether
an a constant
a-decomposition
property
exists
holds
for
such
all bal-
matroids.
7
Conclusions
and
Open
Prob-
lems
DN(e)DN(f)~CU(e)Cu(f)
f #e
f
The
=
~
(DN(t?)@(r?))2
=U
more,
the
if e c CU,
holds
There
and
wefN
thus
= Wf .N holds
exact
are two
dual
obtained
wit h either
series.
from
by
spond
The
als of Ks
and
Ks,s
then
in
fact
be
K4,
through
same
tions;
otherwise,
different
but
the
that
isfying
position.
the conditions
In
a clique
K4
two
one cannot
regulm
cycles
f
in opposite
would
f ) cannot
different
traverse
give
for symmetry
for perfect
e in
sion 1, and hence
and shown
matroids.
There-
of the
decom-
= O if e, f
An
do
above
obtained
efficient
noting
midpoint
f(Z)
to solutions,
for
is the
exambasesma-
cut set expan-
sampling
the role that
and ran-
question
is
=
akZk
the
vertex-pairs
in the
graph)
the proof
~~>~
at
for bases of balanced
(as used
combinatorial
meet
balanced
implies
schemes
intersect
and for bases of
graph
introduced
balance
see [19] and
interesting
pO@IIOdd
36
that
It is worth
at their
expansion;
that
case,
of a graph
We have
A re-
combinatorial
is the
this
is a
by remov-
necessarily
correspond
this
approximation
meeting
all edges
For matroids,
graph.
are ex-
O-1 coordinates).
graph
matchings
[19].
troids,
~ O, sat-
and exact
one has Aef
ple,
corre-
of the polytope.
can be obtained
O-1 vertices
two
and
have
coincide;
is the set
edges
1 (a O-1 polytope
such edges must
graphs
[19]
of a con-
O-1 polytopes
expansion
vertices
where
two
a K4 mi-
whose
). For cert ain structured
the
exchange
all
Vazirani
vertex-set
and
that
complete
midpoint
domized
C.
the
matroid
whose
on O-1 vertices
edges (two
direc-
be nonzero
cycles
horn
problems,
must
find
whose
their
component
= -CU(e)CU(f)Aef
Kn,
du-
a bicon-
as a minor
polytope
graph
cutset
and
I-skeleton
faces (edges)
states
with
other
the
polytope
to l-dimensional
panding,
ing
if it is a series-parallel
traverse
C(e)C(
the
so the
edges e, f that
signs for two
DN(e)DN(f)Ae~
then
furthermore,
graph,
two given
direction
It follows
have
if and only
edge
of the
conjecture
lated
or (2) two edges in
a biconnected
In a series-parallel
cycles
nor.
and
K~
a single
are also excluded,
[28];
do not
namely
as minors,
cent aining
has no K4 as minor
graph.
edges,
be graphic
nect ed component
that
K4 by replacing
are excluded
must
matroids
of Mihail
Define
as the graph
of vertices
decomposition
decomposition,
(1) two parallel
If both
matroid
graphic
or exact
conjecture
unresolved.
vex polytope
123,1. Further-
Wef u =
property
of Aef,
ymmetry
and K~,
fore
~f
following
remains
as well.
satisfys
the
then
the symrnetry
definition
1 = Iq.
UeeCu
u
Similarly,
~
play
in proofs
of Theorem
evaluation
at
definition
Wme
of
2.1.
of
the
point
~,
where
ak
n + k.
is
For
the
number
graphic
the
number
of spanning
the
number
of connected
(1 - p)np~-n~(~-l)
where
pressions
graphic
t ained
The
quantities
eration
for eigenvalues
value
f(1)
gives
plied
subgraphs,
f(z)
reliability
generally
sets (forests
the
of joining
two
property
that
does not
ficients
ak; for
instance,
balance
for
spanning
graphs
hold
for
ex-
case of
the
individual
subgraphs
grateful
comments;
Dyer,
A.
Paul
Seymour
Paul
property.
istair
Sinclair,
Madhu
Peter
Winkler
for
for
coef-
[1o]
to
pointed
out
the nega-
Sudan,
several
to thank
Umesh
valuable
Vazirani,
and
[12]
“On
and
simulated
and
Inf.
[2] D.
Aldous,
“The
of a matroid,
193.
“How
dom?
(On
nent),”
STOC
[4] A.Z.
the
Broder,
trees?
hard
[14]
Eng,
F.
for
trees,”
SIAIW
[15]
the
pp 375–
matroids
CombinatoTial
l?roc.
stmc-
Calgary
In-
1969.
FOCS
1991,
data
structures
and
k smallest
pp 632-641.
On the tree graph
J. Applied
is it
to marry
of
the
Math.
25, pp 187–
M.R.
Jerrum
rapid
mixing
the
approximation
1988,
Jerrum,
perma-
computing
of the
L.G.
Science
M arkov
property
Valiant,
generation
a uniform
L. Lov&sz
D.J .A.
Welsh,
of Jones
Proc.
Carob.
and
Phil.
“Conductance
for
Markov
permanent
and
chains:
resolved,”
pp 235-243,
puter
pp 442-447.
Math.
and A. Sinclair,
the
M.R.
and
complexity
108, pp 35-53.
from
spanning
Vertigan,
polynomials,”
at ran-
[16]
random
D .L.
computational
“Random
pp 50-58.
1989,
ran-
estimating
1989,
functions,
and F. Harary,
Jaeger,
STOC
pp450-465.
“Generating
FOCS
in
construction
labeled
approximation
1986,
simulation
distributions
walk
3, 1990,
“A
for
STOC
connectivity
SIAikl
SOC. 1990,
pp 33–46.
random
Math.
Broder,
chain
Probability
trees and uniform
J. of Disc.
[3] A.Z.
Markov
combinatorial
1, 1987,
Carnegie
discussions.
annealing,”
Sci.
spanning
the
uniform
algorithm
“Ambivalent
2-edge
C. Holtzmann,
Tutte
for
to-
randomized
1992,
Kannan,
applications,
trees,”
and
References
method
R.
bodies,”
Conference,
dynamic
“On
Aldous,
and
polyhedra,”
theiT
Al-
[13]
[1] D.
walks,
and
“Submodular
certain
tures
spanning
insightful
“Random
LPCO
time
of convex
J. Edmonds,
and
in Section
his
We also wish
of Ap-
1991.
zonotopes,
Frieze,
polynomial
for
in particular,
correlation
Frieze,
[11] G .N. Fredrickson,
to
chains ~ Annals
1, 36-61,
algorithm”,
ternational
are
bounds
381.
Acknowledgements
tive
M.
dom
2.
We
Vol.
“Geometric
to appear.
volumes
a
counter-example
connected
[9]
op-
vertex,
simplex
Mellon,
of T [13, 29].
a single
the
see the
dual
under
at
dual
~(z – 1) =
in the
are multiplicative
Strock,
of Markov
Probability,
and A.
[81 M. Dyer
tally
unimodular
sets can be ob-
arguments
D.
and
for fail-
polynomial;
of spanning
by interchanging
and
the
of the last two quan-
Tutte
instead
[7] P. Diaconis
gives
More
for independent
matroids)
size
~(0)
spanning
T is the
of
value
trees,
[27].
sets
the
p; the evaluation
is #P-complete
2’(1, z),
spanning
gives network
ure probability
tities
of
matroids,
and
of combinat
distribution,”
43, 1986,
the
Vazhani,
strut
tures
Theoretical
Com-
pp 169-188.
and M. Simonovis,
chains,
V.V.
orial
“The
mixing
An isoperimetric
inequalit
volume
1990,
,“
FOCS
rate of
y, and
pp
346–
354.
[5] R.L.
Brooks,
W.T.
Tutte,
squares?
C.A.B.
“The
Duke
Smith,
dissection
Math.
A.H.
Stone,
of rectangles
J. 7, 1940,
and
into
[17]
M.
Markov
312-340.
expanders
[6] P.
Dagum
permanent
Journal
and
M.
of graphs
on Computing,
Luby,
with
“Approximating
large
factors,”
“Conduct
Mihail,
chains
),”
(A
ante
and
combinatorial
FOG’S
1989,
convergence
of
Treatment
of
Pp 526-531.
the
SIAM
[18]
M. Mihail
and M.
ties of matroids
to appear.
37
Sudan,
,“ TR-89,
“Connectivity
U.C.
Berkeley.
proper-
[19]
M.
Mihail,
and
U.
Vazirani,
Theory,
[20]
Series
Naddef
[21]
and
ity
in O-1 polyhedra,”
Series
A.
Sinclair,
“Improved
of
Markov
chains
[22] P. Seymour,
[23]
W.
Pulleyblank,
Journal
B 37, 1984,
PTOC. of Latin
expan-
“Hamiltonicof Combinatom”al
pp 41-52.
bounds
and
for
mixing
multicomodit
rates
y
flows?
1992, to appear,
personal
communication.
P. Seymour
and
applications
of an inequalit
chanics,”
the
of Combinatorial
B, to appear.
D.
Theory,
“On
Journal
sion of O-1 polytopes,”
ikiath.
D.J.A.
Proc.
Welsh,
“Combinatorial
y from
C’amb.
statistical
Phil.
Sot.
me-
1975,
77,
pp 485-495.
[24] P. Seymour
and
P. Winkler,
personal
communic-
ation.
[25] A.
Sincl@
counting,
and
M.R.
uniform
Maxkov
chains?
Jerrum,
generation
“Approximate
and rapidly
Information
and
mixing
Computing,
to
appear.
[26]
M.
Sudan,
[27]
L.G.
Valiant,
and
reliability
Computing
[28] D.
Welsh,
personal
“The
y
communication.
complexity
problems
8, 1979,
Matroid
,“
of enumeration
SIAM
Journal
on
pp 410-421.
Theory,
Academic
Press,
1976.
[29] D.J.A.
Notes,
Welsh,
DIMACS
“On
May
the
Tutte
91/Dagstuhl
map,”
June
Lecture
91.
38