Chapter
On
the
Number
of
Milena
18
Eulerian
Orientations
MihaiP
Peter
Abstract
orientation
We give efficient randomized schemes to sample and approximately count Eulerian orientations of any Eulerian
graph. Eulerian orientations are natural flow-like structures, and Welsh has pointed out that computing their
number (i) corresponds to evaluating the Tutte polynomial at the point (O, –2) [8,19] and (ii) is equivalent to
evaluating “ice-t ype partition functions” in statistical
physics [20].
Our algorithms are based on a reduction to sampling and approximately counting perfect matchings for
a class of graphs for which the methods of Broder [3,10]
and others [4,6] apply. A crucial step of the reduction
is the “Monotonicity
Lemma” (Lemma 3.3) which is
of independent combinatorial interest. Roughly speaking, the Monot onicit y Lemma establishes the intuitive
fact that “increasing the number of constraints applied
on a flow problem can only decrease the number of
solutions”.
In turn, the proof of the lemma involves
a new decomposition technique which decouples problematically overlapping structures (a recurrent obstacle in handling large combinatorial populations) and allows detailed enumeration arguments. As a byproduct,
(i) we exhibit a class of graphs for which perfect and
near-perfect mat things are polynomially
related, and
hence the permanent can be approximated, for reasons
other than “short augmenting paths” (previously the
only known approach); and (ii) we obtain a further direct sampling scheme for Eulerian orientations which is
faster than the one suggested by the reduction to perfect
mat things.
Finally, with respect to our approximate counting
algorithm, we give the complementary hardness result,
namely, that counting exactly Eulerian orientations
is #P-complete,
and provide some connections with
Eulerian tours.
such
that
this
Research,
445
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South
Street,
close
uniform)
to
orientations
ining
counting
tion
function
is equal
orientations
of some
underlying
ther
see also
[2,14]).
or hardness
ZICE,
for
one
ing
scheme
the
first
our
#P-completeness
result
lack
of any formula
or other
ZICE
for
efficient
exactly.
to perfect
the
Ising
type
ing
(The
list
monomer-dimer
model,
model
which
percolation
walks,
remain
In addition,
counting
for
approximation
plane.
to
ations,
that
trees
reliability,
the topology
in [12];
and
the
problems
and
ice-
involv-
self-avoiding
on
and
the
map
problems
the
that
Tut t e plane
forests,
configuration
nodes,
preof the
corre-
are,
colorings,
model,
point
the
((), –2),
results
on the
of a net work,
through
that
corresponds
at the point
Ising
practical
[8,19]
hardness
translation
is a uni-directional
flows
directly
in [3,9,10,18];
orientations
and
curves
one further
preserves
the
to compute
contains
treated
fundamental
or
spanning
explains
translates
treated
polynomial
a direct
Other
points
is
while
problems
also observed
Eulerian
the Tutte
have
5.1)
polyominoes,
therefore
here
count3.4)
open.)
to evaluating
sented
answer-
ZICE,
method
which
it has been
problem
the
(Theorem
here;
fur-
evaluating
(Theorem
of Welsh’s
theory,
which
for
Thus,
approximate
has been
(for
no reasonable
approximate
efficient
is treated
parti-
of Eulerian
graph
graphs.
and has been
which
number
our
problem
matchings
observed
crucial
was known
to
of exam-
Welsh
the
orientations
scheme
of Eulerian
problems
Eulerian
questions,
Eulerian
Eulerian
of some
Apparently,
planar
ing
the
of Welsh’s
to the
result
4-regular
of
in the context
model”,
“ZICE”
algorithmic
set
In particular,
“ice-type
of
arbitrarily
the number
by Welsh
[20].
questions
graph.
complexity
physics
even
the
out
finding
efficiently;
the
Eulerian
computational
in the so-called
that
a distribution
of counting
was raised
details
with
from
and
that
tation
138
generation
in statistical
From
e.com
concerned
edges
directed
known
can be accomplished
are
significance
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of its
of edges directed
of edges
is well
of an arbitrary
orientations
orient
Morris-
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orientation
number
It
orientation
(i.e.
The
to the
dOUt (v).
sampling
represents
*Bell Communications
town, NJ 07960.
mihaiI@Hkh.bellc ore.com
=
paper
example,
Consider an undirected Eulerian graph, that is, a graph
in which all vertices have even degree. An Eulerian
is an
v the number
an Eulerian
in
sp ond
Int reduction
graph
vertex
din(v)
Tutte
1
of the
for every
v is equal
v:
a Graph
Winkler*
towards
of
of
for
acyclic
etc.
of view,
if a graph
an Eulerian
of the
thus,
oriennetwork
a maximum
EULERIAN
global
ORIENTATIONS
flow
sampling
without
inst ante
number
of Eulerian
ing the
of such
number
category
but
paradigm
suits,
concerns
ber of perfect
paper
the
obtaining
equivalence
and
sampling
and
Vazirani
Broder
and
imately
counting
Markov
chain
also
[4,6] for
further
for
schemes
input
which
run
graph,
accuracy,
and
perfect
over
graph
bilities,
can
in time
ratio
perfect
while
ing
edges.
#P
of the
count-
the
lM._
known
and
In
Section
size of the
1 l/lMn
small
I of the
failure
in-
proba-
3 we show
pling
and
a class
graphs
obtain
t ations.
It
on I&tn_l
treated
previously
pander
graphs
arising
from
by showing
for
which
[ for
(dense
[10], graphs
that
perfect
remark
classes
with
sequences
augmenting
path
graphs,
log n length
for
(e.g.
expander
=
that
large
[11]
random
factors
etc.)
were
matching
constant
graphs
to sam-
length
etc.)
for
0(n4),
Eulerian
of graphs
[3],
for any near-perfect
“short”
for
graphs
approxi-
matchings
lMn_ll/lMnl
to
l/lM.
degree
and
reduces
solutions
interesting
bounds
sampling
counting
efficient
is
that
orientations
approximately
of
we
Eulerian
orien-
all
known
that
were
and
ex-
[4], graphs
this
large
way
for
Markov
of
(hence
the
cuts,
to
may
random
rapidly
an
tions
into
paths
mixing.
efficient
matching~.
and
Euler
Using
sampling
a polynomial
no
of
In
scheme
factor
over
In Section
5 we show
exactly
is #P-complete,
that
has no
arguments
paths
on
the
orientations
facilitates
the
Hence,
Eulerian
indirect
counting
thus
that
4 we
technique
the simulation
for
to
and
Section
our
congestion.
directly
paths
chain
applications
such
how
transition
many
an “expander”
y in
chain MC
states
introduction
number
elements
without
an
is to
between
too
further
outline
one
property
populations
edge.
Markov
graphs
related
example,
on such
the
de-
structures
that
for
difficult
transition
of ver-
which
mixing”
with
walk
we
a technique
For
such
[10]
the
pair
of the Markov
see
in
graphs
cardinalities
argue
bound
same
decomposition
of
dense
any
the
Eulerian
there
Thus
and
define a natural
of
and
ver-
paths.
of paths
[6,4, 12] for
Now
decouple
use
on
that,
two
difficulty
any
“rapid
graph
and
like
elements”
congested
fast;
outgo-
overlapping
and a random
reasoning,
saves
is a
is
converges
improvements).
is
“Euler
number
chain,
chain
incom-
we introduce
defined
the underlying
“small”
graph
between
crucial
to orient
than
arbitrary
populations.
chains
Markov
of
in
in handling
the
outgo-
than
The
overlapping
a substantial
the
behave
problematically
establish
both
3.3 we show
edges.
into
combinatorial
to
incoming
graphs
paths
obstacle
and
ways
() d–1
more outgoing
problem,
graphs
two
edges.
so that
of incoming
is that
problematically
Decoupling
to
a fact
disjoint
obtained
for
parallel
isolate
is a recurrent
edge
of near-
with
of so-
is when
2d parallel
Lemma
all
favor-
number
the edges
more
locally,
To bypass
couples
two
solutions
constraints
case
with
re-
2d
Monotonicity
of decomposing
this
counting
In
u has two
of
the
simplest
are only
larger
to the car-
hence
decrease
number
v has
on
in our
particularly
number
to orient
same
such
consider
approximation
numbers
exponentially
(see
randomized
in the
cannot
novel
sampling
via
total
of
approx-
the
that
ways
there
in some sense,
tices connected
tices.
Valiant,
[7, 11,12, 15] for
desired
matchings
with
so that
edges,
sampling
be achieved
of the
u and
() d
v have the
edges
efforts
for
specifically,
suggest
are
the
class
using
polynomial
inverse
the
(and
More
2d
There
but
only
connected
ing
scheme
[10]
and
can
based
much
arise
are related
the
v are
u and
I is not
that
exhibit
the
num-
as usual).
mately
thus
the
the
for
which
example,
can be
seminal
Jerrum,
matchings
matchings
counting
re-
the
concrete
improvements,
For
polynomithe bound.
I is not
sets of orientations,
problem
establishing
Sinclair
technique
results).
perfect
that
lMn_I
of graphs
“increasing
edges,
approximate
the
a flow
vertices
why
problems,
behavior:
1I/ lMn
cardinalities
of certain
only
yields
argument.
class
these
ing
directed
by
by
and
perfect
various
approximate
for
result
is that
more
of
the
with
which
on lMn–
path
the reason
[ for
finalities
able
bound
of bipartite
in Valiant’s
established
followed
[M.
to flow-like
in
approximations.
simulation
relevant
results
hardness
Jerrum
than
lutions’).
one
our
augmenting
even
hardness
counting
of randomized
[13],
[3]
the
matching
but
is
collapses
and
to be complete
was
In contrast,
outstanding
is also
waa shown
efficient
The
significantly
a perfect
of
problems
mat things
time,
latter
matchings,
Roughly,
in the
solutions,
the most
which
that
matchings
In turn,
towards
put
known
on permanents
[18].
harder
algorithmic
in polynomial
falls
unexpected
the case of perfect
it is well
constructed
can be associated
a short
of max-
a solution
set of all
Perhaps
our
matching
few near-perfect
network.
question
constructing
are either
behavior,
both
the
orientations
unless
occur.
of such
use for
graphs:
ing
sense,
classes
we shall
of view,
the
or intractable,
complexity
the
the
count-
number
the apparently
countin$
in an approximation
involved,
to
ally
duction,
for which
time,
and
counting
around
total
Eulerian
of problems
of sampling
flows
point
counting
in polynomial
while
larger
perfect
a
behaved.
a theoretical
and
Consequently
to observing
is equivalent
of maximum
are better
From
sampling
sinks.
a network,
with
139
amounts
orientations
networks
flows
and
orientations
random
imum
sources
Eulerian
Presumably,
OF A GRAPH
choice
MC
of A4C
is
as
orientations
reduction
Eulerian
justifying
of
to
orientathe
neces-
MIHAIL
140
sit y of our
well
efiicient
aa the lack
counting
method.
positions
into
The
the
known
some
terms
3
we
a class
Section
can
the
chain
defined
faster
sampling
such
an approach
how
the
tours.
rapid
mixing
directly
on
result
and
Summary
property
We
faster
5
open
and
the
the
a connection
and
a
In
Markov
ratio
potential
of
counting
lMnJ–l
for
algorithm
numbers
Let
with
probability
outputs
scheme
for
y at least
for
It$l for
aim
is
to
obtain
approximation
tations
efficient,
c-l,
logd–l,
Let
IV(I
of perfect
graph
running
log6-1,
=
Let
matchings
respectively
be
algorithm
that
●
and
where
and
near-perfect
matching
IVI = n.
in
n,
where
be
the
[10],
there
in n’,
The
log$-l,
matchings
of G’
is a mat thing
2.2.
set
of perfect
scheme
in
is
an
runs
the
[?]
and
For
scheme
[Broder
there
Jerrum
[6].]
and
improvements
as above,
the
in
M.,.
logd-l,
FACT
[3],
is a d-sampling
matchings
[10],
[Broder
improvements
above,
I{u
– ~
any
for
in
ratio
For
[4,6].]
of
matchings
M~J.
Sinclair
<
V-
={v
the
last
graph
G’
●
if
out
Eulerian,
regard
{u, v}
P as
c E
towards
then
v),
or
of v).
and
a vertex
v c V,
the
: (U, V) E F}l
for
~,
where
d(v)
zero
charge
is called
of a positively
directed
P=
degree
balanced.
vertex
of v that
must
be
balanced.
(V, F),
let
: q(v) <O}.
is the
charged
out
v to become
orientations
V+=
The
{V E V : q(v)>
of P is
charge
graph
scheme
where
?k
=
G’
runs
for
1)I?%-11.
the
ratio
k
there
verify
from
first
matchings,
one
:dP=k}
>
IE I/2.
we
near-Eulerian.
some
in V+
sense,
expect
Eulerian
This
lMn/_
behavior
last
orientations
bound
1 l/ lMn~
upon
the
I for
for
any
cardinalities
and
k,
that
3.3
show
we
that
l~k I ~ n(n
is polynomially
graphs
“more
?k _ 1, and
Lemma
generality,
and
flows
in
Monotonicity
G,
flow
that
in V–.
in ‘Pk are
in full
graph
by
to verify
one set of 2k edge-disjoint
to reflect
In
hard
to vertices
than
this
~h_~.
this
Furthermore,
orientations
constrained))
,
it is not
is at least
vertices
might
any
to
charge
considerations,
from
In
easy
near-perfect
with
0 for
preservation
formalize
for
is
to
let
Of Qk and
Sinclair
scheme
equality
analogy
P~ ={P
one
any
charge
EV
By
Finally,
as
polynomial
and
<
with
of edges
in order
call
time
The
may
is directed
● F}l-l{u
q(v)
the
principles.
paths
lM~J_ll/lM~~l.
For
we need
necessarily
We
where
edge
: (v, u)
number
where
severely
(~, b)-approximation
not
P = (V, F)
an orientation
the set of perfect
, Jerrum
cardinalities
terminology.
is directed
A vertex
for any P G ~k
2.1.
(the
sets
size n’–l).
FACT
the
reduction,
(c, 6)-
graph,
Mnl-l
(a near-perfect
in the
P = (V, F),
edge
example,
shall
and
problems.
about
orien-
polynomial
a bipartite
M.,
by a
efficient
of v is
reversed
as usual.
V~, E’)
2n’.
(V, E),
2.2 imply
2
For
S is an
times
which
above
outputs
set ‘PO of Eulerian
G=
arise
S is an
1 –&,
d-sampling
the
Eulerian
= (V{,
lV~l
for
we mean
and
G’
=
eficient
schemes
of any
By
(the
of v in G.
ISI such
2.1 and
argue
countfor
bounded
G = (V, E).
an orientation
v is the
scheme
of graphs
an orientation,
graph
graph
Clearly,
an estimate
approximately
orientation
to
further
(u, v) E F
O}, and
Our
some
a directed
c For
6.
that
(c, &)-approximation
approxi-
of Euler
~lpr[s=z]-&l<d
An
and
and
Facts
– 1 that
P denote
either
XEs
that
M.)
~(v)=
that,
ratio
is polynomial-
a class
is always
Eulerian
of an Eulerian
Preliminaries
a s G S such
for
in order
to introduce
charge
2
the
sampling
orientations
l/lMn,l
the
and
that
sampling
in n’, hence
(v, u) E F
are in Section
For some set S # 0, a d-sampling
to
mat things
However,
●
a
complementary
with
problems
the
of A4n,
hence
approximate
contains
and
technique
for
discuss
perfect
schemes
matchings
decomposition
f– 1, logd–l,
n’,
of Eulerian
reducible
ing
to
efficiently.
orientations,
to yield
sampling
perfect
cen be handled
Section
hardness
counting
scheme.
algorithms.
from
in
3 we show
counting
polynomial
orientations
that
4 we outline
and
of orientations.
Eulerian
Section
mate
follows:
matchings,
reduction
counting
of graphs
yield
for
as
polynomial
In
a connection
organized
results
and approximately
time
WINKLER
lA4nJ-11/lA4mIl.
time
is
the
in
of decom-
we observe
for the treatment
give
approximately
application
as
exact
tours.
2 reviews
sampling
for
of
e counting,
reasonable
one further
paper
Section
approximate
elements
the
introduces
and
for
of Eulerian
rest
Section
By
Euler
wit h numbers
In
scheme
of any computationally
AND
arise
–
‘related
to
from
the
EULERIAN
ORIENTATIONS
reduction
of counting
perfect
mat things
efficient
sampling
Eulerian
In this
and
section
imate
counting
of any
and
[E I = m.
Let
The
=
where
Yv
{y~,i
and
v;
=
fiecE{w.}
, 1 <
i
=
each partition
any
Eulerian
(
bY I-I.ev
PROOF.
can
class
G
as
is
a
pair
exactly
one
cover
G,
first
that
with
each
so
with
and
!.
of
divide
each
must
matching
Eulerian
edge
{xv, ~, we }
of which
be in
of
{Zu,e,
the
of G’
orientation
{u, v}
and
direct
notice
be
that
matched
forces
the
G,
we}
perfect
orientation
there
in
G’,
fact,
11.<v
that
(~)!,
balances
the
each
of
partition
‘ince
‘rice
that
XV
and
(~)
3.3.
thaf
Let
gives
class
has
‘he
‘e’s
perfect
rise
each
that
cardinality
‘atched
of G,
and
matchings
2.1 and 2.2
of Eulerian
be upperbounded
by a
reasoning
lM~~-11
! 1711,
as in
< 2~lM~/1+
and
the
proof
follows
❑
fullpaper).
where
Lemma.)
For
any
IVI = n,
and for
any
lE1/2,
q be a realizable
vertex
exists
.
q(v).
Clearly
and
V-,
respect
decreasing
by
one
We
~~1~
shall
exactly
‘ith
q is fixed,
and
q.
v c V-,
show
and
that
leaving
the
K%(!7)I
s
I
be
the same
V+
the
few
next
V+
let
u E V+,
such
%(q)
v has charge
and
V-
Q~ be the
can be obtained
some
from
increasing
q by
q(v)
q otherwise
the
holds
any
following
‘:
e
Finally,
for
%-l(q’)1
u
q’cQ~
To prove (1) we shall relate orientations
orientations
number
each
of
in
orient
ation
orientations
tion
Ug,e ~, ~fi-l(q’)
of orientations
in
in
~k (q)
is
~k (q)
in
Ug, c Q, ?k_l(q’).
relate
orientations
and
in ~fi (q) to
so that,
roughly,
in Uq,c Qq Tk_ 1(q’)
of
in
is by
v E V-.
what
related
than
with
course,
Tk (q)
reversing
In
larger
related
a path
follows
sition
in Euler
elements
the
with
the
number
each
orienta-
the
with
called
“Euler
elements”
so that the
be reversed
can be e~ily
accounted
a nonand
For
that
by one for
some
Let
q(v).
fixed
functions
q(v).
in ‘Pk (q) have
that
on V,
orientation
each vertex
k = ~Ucv+
to the
q(u)
for
same.
clearly
function”
one
charge
such that
we assume
with
least
all orientations
and
paragraphs
are
“charge
at
v 6 V haa
the set of orientations
u c V+
re~oning
to
G,
natural
waY
orientations
in
with
we
endpoints
introduce
into
techniqueto decompose an Eulerianorientation
Eulerian
mat things,
are
can
(Monotonicity
there
U g, ~ Q, Tk_ l(q’)
with.
perfect
orientations
and near-perfect
G = (V, E),
l<k<
suppose
to
outgoing
to the
to see by similar
orientation
XV,
associated
Hence
according
in YV
in
in
their
incoming
are associated
class
vertices
orientation.
it is easy
Eulerian
to
is indeed
vertices
vertices
matched
corresponding
partition
~
d(v)
~
be
they
v, all
of the
can be partitioned
that
each
each
a unique
this
•l
graph
straightforward
forthe
graph
PROOF.
i.e.
mat thing
Thus
To see that
~
to
Furthermore,
empty
as (v, u).
remaining
clearly
of the
mat things
that
v}
for
to
unmatched
we ‘s, which
edges
{u,
of G is obtained.
Eulerian,
which
but
we.
orientation
remain
LEMMA
that
approx-
of G’
of
YV ‘s.
set
~
()
number
efficiently
perfect
a unique
For
edges
otherwise
must
tedious
are left
k such
Further-
the
ways
!
the
, where PO and T,
near-Eulerian
of G if I?l l/l’Pol
in n.]
By
(details
the
the
G.
~V~V
approximate
matchings
4jQ
with
Eulerian
are perfect
-1- 1)2 ~ucv
Eulerian
If {XV,., w,} is in the perfect matching
of G’ then direct {u, v} as (u, v) in the orientation of
to
and
set of charge
follows:
of
(%Y
=
hence
G,
of
of G if we can
XV’s
any
For
3.1, it can be shown
x Y..
relation
has cardinality
of perfect
associated
of
E’
x.
!$!l)1!$
Note
be
=
of G’ can be partitioned
orientations
of the
1~~1< n(n - l)lp~-11
graph
are in one-to-one
efficiently
the number
edges
= 2m,
are
= O (n21?~\/\%l)
in the size of G.
orientations
imate
d(v)
there
with
Xv
Uvg”
half
3.2.
orientations
polynomial
PROOF.
U (JVEVYU,
u
LEMMA
Lemma
and
‘we }}
IVJI = ~VeV
classes
can
~},
<
We}, {Z”,.,
the set PO of Eulerian
Eulerian
n
to a perfect
graph
V,
of G’.
[Hence,
by Lemma
3.1 and Facts
we ~an efic2entlY
Upproxamate
the number
[VI=
where
each
other
are Eulerian
orien-
where
be a bipartite
“ Xv,
the
MO I and MmI_l
of Eulerian
problem
for
[A4~-1[/lM~~l
and approx-
(V, E),
of this
of perfect matchings
we
G=
Uv
{{ZU,~,
the partition
more
for
results.
sampling
Xu’s
()
Counting
=
e},
n’ = IV(I
[Hence,
approximation
V(
LEMMA 3.1. For
that
schemes
of the
matching
graph.
the set P.
V;, E’)
size of G1 is polynomial
M~I
half
we obtain
is as follows:
(V{,
=
Ue~E:e={u,u}
Clearly
for
graph
bipartition
UeCE:e={u,v}{~v,
the
reduction
problem
G’
vertex
Eulerian
efficient
schemes
Eulerian
to counting
thus
counting
Approximate
we present
tation
mat things
of any
to establish
Sampling
orientations
3.1 and 3.2),
and approximate
We proceed
3
Eulerian
(Lemmas
orientations
141
OF A GRAPH
a
so-
number
of paths to
for. The decompo-
is as follows:
in
(I).
A circuit
(w1, w~),
(w’,
of an orientation
w~),
. . .. (wl_l,
P=
Wl),
(V, F)
is a sequence
(WI) Wl),
where
all
142
MIHAIL
the
(wi,
(wl,
Wi+l)’s
are distinct
WJ2), (wz, WI)}
as (~z,
W),
(W,
a generalized
(Is).
A circuit
(lb).
(w3,
A circuit
in F.
there
(w,
is no starting
circuit
that,
i.e.,
w2),
e.g.
circuit
there
are ~W ~vlv+uv.
~ for
a circuit
is
each
Now
point.]
Pin
~
()
U Ug,cQ,
‘P~(q)
to prove
! distinct
WINKLER
sets of pairings
‘P~-1(/).
(1) it suffices
to show
if no edge is incident
(w,vguv-(%)!)l~k(q),~
of P is a V+-circuit
one incoming
[Note
is the same
U V-.
in the
to a vertex
w),
of P is a free
in V+
edges
edges
W3), (W3, Wl)
w3),
cycle:
to a vertex
two
(wl,
AND
circuit
if there
incident
and
one outgoing,
in V-.
Analogously,
to some
and
(wJ!uv_(%)
,,y’,-(q’),
are exactly
vertex
in V+:
no edge is incident
define
V--circuits.
Consider
~wEv\v+uv-
~
copies
!
of each
ori-
()
(II).
A
(V+,
(w~,lo2),
(tom,
w3),
and
v),
moreover,
as in
V-)-path
(w~,
all the
circuits,
has
point:
(V-,
(III).
a V+-circuit,
a (V–,
v),
all
point:
u,
V+)-paths,
or
or
or a (V+,
entation
of pairing
wi’s
in F;
(which,
are
tained
)-
in
a free
a (V+,
circuit,
or
V-)-path,
V+)-path,
or
or a (V–,
V–)-
to
V+)-paths,
lated
they
P = p’,
that
U~~~Q, %-1(~’)
tinct
each orientation
for
of the
~
F)
haa exactly nW6v\V+Uv.
partitionings
see this,
P = (~
of its
each
edges
vertex
into
~
()
elements.
Euler
U V–
w c V \ V+
c ‘P~(q)U
! disTo
consider
each
(P’,
Now
k >1
counting
completes
! pairings
()
in w.
edges
pairing.
For
The
suggested
of the
some
vertex
partition
by some
incoming
w,
of F into
fixed
let
and
q = {pW
elements
: w E V \ V+
proof
p’)
are re-
that
and
is related
by
related
a (V-,
to 2k+c
is related
Vf)-
(P, p) ’s,
to at most
(2)
holds
c+l
true,
be-
by elementary
and
hence
(1)
also holds
step
where
we
allow
of Lemma
is
in
suggested
path
to see that
It
true.
q to vary
3.3:
outgoing
pW denote
Euler
the
)-paths
also be c
(P’,
2k + c ~ c+ 1, and
final
V-
must
and
in P’
~’)
P’),
V– )-paths
there
reversed
(P, p)’s
hence
following
then
if (P, p)
(P’,
@ = p.
p’,
V-)-path
if (P’,
it is easy
and
(V+,
moreover
2k + c (V+,
one
considerations,
The
one
is a (V+,
Hence
p’,
to
decompositions
the
one of these
~’)’s.
cause
for
orientation.
if P can be ob-
are
that
identical
except
in P.
then
We argue
have
which
by
and
only
one of the
by
if there
are suggested
(V-,
them
by reversing
each waY P
of the
is related
if and
are suggested
for
one
edges
P E %(q),
T,_l(q’),
see that
that
~k–l(q’)j
the
where
P’
that
path
path.
(P, p),
from
P’
P’
U.Uq,cQ~
partitioning
P’ E Ug,~Q,
easy
V+ )-paths,
and
that
in
V-
an ending
(V+,
Say
in ?,(q)
where
not
a (V+,
and
WJJ,
defined.
of P is either
a V--circuit,
V+)-path,
(u, Wl),
(u,
edges
the
distinct)
are analogously
element
where
to circuits,
starting
(V-,
sequence
are distinct
and
necessarily
Finally,
Euler
a
(w,
in contrast
V-)-paths
An
is
w),
W;+l)’s
not
that,
a specified
v).]
and
P
v c V–,
are
[Note
path
(wi,
u E V+,
V+UV-.
of
. . . . (w-l,
such
a
that
is
U V– } is
as follows:
.
Free
circuits
are
$7W2(W,
W2))
PW(W~,
w1)(w1,
●
V+-circuits
W1),. ...
and the
wi’s
E V\
are of analogous
●
(V+,
the
W)
form:
=
of
(w,,
PW, (W1,
WZ), where
are
PW, (W,
of
(W>
...,
the
the
W2), (W2, w~)
=
W),
=
(W,
Wi’S E V \ V+
form:
Wz)
U V-.
(u, WI)) (WI,
(w~, ~) = P~,(w/-l,
V+ U V-.
And,
W2)
=
w~), where
u c V+
of course, V–-circuits
form.
V-)-paths
where
are
of
the
form:
(u, WI),
(WI,
W2)
. . . , (w1, V) = PW (w1-1, wl),
= Pwl(%wl),
where
the
u~V+,
VGV–,
and the wi’s ~ V\
V+ UV–.
And,
clearly,
(V–,
of analogous
It
of
F
of pairings
(V+,
V+),
and
(V–,
V–)-paths
are
form.
is not
into
V+),
hard
Euler
•1
THEOREM
IVI
to see that
q suggest
different
this
and
is indeed
that
partitions.
two
a partition
distinct
Furthermore,
sets
n,
For
there
approximate
counting
orientations
of G.
n, E‘1,
logd–l,
PROOF.
2.1 and
elements,
=
3.1.
(d, $)-sampling
schemes
The
for
G=
(V, E),
and
(c, S)-
the set ‘PO of Eulerian
running
time
is polynomial
in
Follows
by Lemmas
3.1, 3.2, an 3.3, and
Facts
❑
2.2.
REMARK.
We prove
of Theorem
is needed.
are
graph
and log8-1.
the
proof
any Eulerian
In
the
Lemma
3.4
following
3.3 for
only
the
section
general
ii, while
special
case
we need
both
k =
in
1
cases
EULERIAN
ORIENTATIONS
k =
1 and
that
estimates
k =
reduction
full
2,
and
in
to perfect
3 at the
end
general
rather
matchings
We shall
143
a potential
ITO I directly
generality.
Remark
OF A GRAPH
than
Lemma
have
more
of Section
scheme
through
Let
the
3.3 is needed
to say about
this
with
PA=
q(u)
in
ence”
in
symmetric
4.
PAB
V\{u,
Euler
Decompositions
Arguments
Theorem
3.4 suggests
by simulating
latter
UMnJ_
Mnl
property
(this
attention;
etc.
obtain
a direct
rapidly
mixing
the
is the
elements
for
so that
bounded
[10].
the
the
by the
Sinclair
“path
chain
on
received
[1,3,10,16,17]
we
outline
how
PO,
by simulating
to
a
on ‘POU’P1. Except
is rather
use
obvious)
for
the
new
of decompositions
definition
of
“random
path
encoding”
method
chain
can
of Jerrum
probabilities
on state
defined
be
and
U ‘PI
7’o
as follows:
be the state
= (V, F(t))
space
on time
If “heads”
i!;
Pick (u, v) E F(t)
uniformly
F(t) \ {(u, v)}
u {(v,
u)})
G 70 UP1
action
circuits
and
to another
have
is
thus
to
move
walking
by reversing
and
to
the
check
aperiodic,
distribution
involved
unit
charges
from
edges
that
and
one
Define
the
unit
to show
MC
that
with
from
a ({u},
on
the
therefore
it converges
U ?1.
It
irre-
to the uni-
is significantly
over
PO
that
MC has the rapid
mixing
prop-
THEOREM
4.1.
For
any
Eulerian
graph
G and
any
<
2°(m)e-o(*)
(Outline).
of Eulerian
we shall
bound
edge.
standard
This,
path
in
arguments
turn,
and
{u}-
v.
by
its
{v})-
restriction
reversing
for
{v}-circuits,
a free
circuit.
its
from
u. A ({v},
edges
successively
So a ~A&canonical
c
P.
edges
{u})one
path
of steps
1) !c! distinct
its
a (u, u’)
reversing
+
as
then
starting
from
for
of a sequence
c’!(c
and
by
edge
PB
of ({u},
no other
edge
starting
to
order);
from
on states
PAB-canOnical
pair
PA
from
uniformly
at
path
P~/)
random,
uniformly
path
E(PM, &)
<
bound
and
on
d(t)
with
of pairs
from
the
6
PI,
by first
pick
a
choosing
then
chosing
a
at random.
number
along
combined
PB
to PB
Let E(P~,
be an edge in MC.
the expected
It can be argued
and
PA
(PA,
PB)
Pjwl)
such
PA to PB uses (PM,
lines
[Po
of the
a bound
of [4,5]
U
that
PMJ ).
that
pll~(~4) ,
theorem
in
follows
by
the
[6]:
.
We shall
any pair
{u})-
-paths,
(v, v’)
reversed
indeed
path
(PM,
the
c ({v},
PA
starting
first
from
representative
and
=
PA to PB.
pAB
above,
pAB
circuits,
any
reverses
with
by reversing
are
each
t:
pAB(W)’S,
from
is reversed
a
is
starting
from
denote
edges
the terms
and
induces
a decom-
(in
{u})
(and
one at a time
There
Let
({v},
circuit
{v})-path
~AB-CanOniCal
more
let
and
outgoing
free
path
one at a time,
is reversed
some
let
distinct
circuits
lexicographically
For
charges
of
{u]-circuits,
({u},
w in PAB,
of w,
{v})-paths,
by alternating
A
for
the
incomincoming
pAB ‘s, where
include
the
{V})-PATH
order).
edge
the
PROOF.
paths
successively
paths
({u},
which
reverses
edges
path
c+l
reverses
the
paths
ation
is symmetric,
with
dAB(w)!
a “~AB-canonical”
first
reverses
along
orient
one more
vertex
distinct
u must
than
{v}-circuits.
PA to PB consists
ert y:
time
circuits,
in
c’ circuits
“representative”
is trivial
ducible,
and
of MC.
traversed.
It
form
MC
of
paths,
clearly
d AB (w)!
of PAB
paths,
at a time
then P(i! + 1) :=P’
else P(i! + 1):= P(t);
So the
position
successively
at random;
have
in
(same
while
outgoing
of incoming
(B denotes
all vertices
edges);
more
“differ-
are balanced
{PAB(W)
: w C V \ {u, v}}.
Following
the reasoning
of Lemma
3.3, each pAB
A
else
(where
in-degree=out-degree
are
HWev\{U,UI
and
then P(t + 1):= P(t)
lJ’=(v,
be a pairing
one that
canonical
the
There
outgoing
c PI
the
in PAB
For a balanced
pAB
w.
PB = (V, FB)
Consider
that
v must
edge.
denote
(w)
let
and they
one
similarly
outgoing
into
congestion
and
and
with
dA~ (w)
and
much
even degrees
degree
edge,
and
= —1.
Notice
of incoming
odd
than
in
mixing
fair coin;
Toss a
If
proposed
In particular,
transition
Let P(t)
for
MC
expected
MC be a Markov
Let
and
(which
here
have
set To
rapid
is referred
scheme
chain
the
Mm. As
now
section
of MC
point
paths”,
this
sampling
Path
is a Markov
by
reader
In
for
the so-called
has
Markov
definition
technical
Euler
property
details).
for
scheme
possesses
the unfamiliar
for
scheme
scheme
sampling
I which
further
q(v)
: PAB = (V, FA @ FB \ FA)
v} have
ing
a sampling
a sampling
in [3], the
and
~ ?.,
and
difference).
number
4
(V, FA)
= +1
near
congestion
implies
[4,5,6,10].
first
define
Eulerian
through
rapid
paths
beween
orientations
any
mixing
and
particular
following
where
d-1
is the transition
is rn- 1, 1 is the
our
length
case is m. and
probability
of the
c is E(PM.
y which
canonical
PMI).
paths
in our
which
case
in
❑
144
MIHAIL
REMARK
by
1. A sampling
simulating
ciently
MC
small,
claim
fail
tions
yields
scheme
for
and if I’(to
and
P(to)
so that
) E 70 output
repeat.
Roughly,
E 7’0,
the
d(to)
is suffi-
P(to),
one
and
oriented
for TO can be obtained
to steps,
out
otherwise
of nz simula-
whole
scheme
is effi-
receives
receive
exactly
d(v)–1
of the edges
out
REMARK
2.
reduction
to perfect
Broder’s
The
Markov
of Section
sampling
only
for
through
‘Po
is obtained
the
by simulating
on A4nf UA4n/-1
[3] for the graph
The
convergence
of this
and,
since
MC
reduction
to perfect
REMARK
3.
chain
=
yields
mat thing.
for sampling
mat things
a perfect
To is faster
by a factor
O(n’4),
than
defined
these
for
~~
chains
= l$=op~
analogous
for
are rapidly
approximate
chains
chains
all
mixing,
estimates
1<
rl &
by (i)
lPo1/1~11,
m
‘111/2 ~
the
ri’s
l~IEI/2-1[/l@I121,
are fl(n2))
(ii)
s to V1 are oriented
out
V2 to t are oriented
towards
Xj
VI
Eulerian)
are
PROOF.
We
IVII
=
IV21 =
perfect
that
n,
argue
=
of
lMn
that
(Vl,
IEI
matchings
computing
we
a reduction
G =
oriented
m,
and
G.
It
has
with
lMn
that
counts
Eulerian
orientations.
Assume
without
loss of generality
at least
a graph
E U E$ U Et,
edges
VI,
so that
parallel
edges
connect
Hence
are
~V~v,
the
graph
Gm,
connecting
together
with
k parallel
Now
realize
that
edges
symmetry
l~kl
(5.2)
t to s, all edges connecting
edges
edges
between
ml
edges
t. We have
lMnl
implies
that
vertex
t},
and
d(v)
for
edges.
– 2~ =
together
A’j
s and
with
m’
where
if m’
is odd,
.
set of Eulerian
. . . . m’
then
if
the
in terms
m!
orienta-
is even,
cardinalities
of the
and
of the
cardinalities
of
follows:
as
be oriented
t must
(5) holds
be
since
i parallel
edges
remaining
k<” from
that
()
ways
between
V1 to be directed
from
there
k
are
i
of directing
s to t and the
s and ‘tfrbm
t to s, and this forces exactly ~a”
between
s and
s to VI
(so
s is balanced).
calls
to
(one
the
an
query
(5)
yield
l~k,
for
Finally
tours:
graph.
Let
v into
P
=
each
{$%
counts
graph
and
time)
lMml,
from
system
(3),
(4),
in
and
unknowns
lMml
can be
can be solved
cl
the following
orient
Let
G=
(V, E)
p.
be
a partition
:VE
which
the
lm–2nj+l
orientations
.
of Eulerian
pairs
hence
equations
is easy to see that
by
Eulerian
G,),
2 independent
we provide
number
to
I‘s can be inferred
that
. . .,m’,
(it
in polynomial
Euler
l’Po(G~)
oracle
M’+
k=O,
deduced
the
of G~t
t are directed
V2 to
‘S
the
k = 0,2,
can be written
(Gk)’s
Gk is G’
s to t.
orientation
denotes
where
., m’
Finally
– 2
t to V2. Finally
in general
if ‘PO(Gk)
k=l,3..
m’~k>[~~
l~m,-kl
in
each
similarly
s to V1 must
connecting
of the
towards
of
oracle
are
and
=
of G~,
connecting
there
connecting
in an Eulerian
that
all
is G!
j
V2
connecting
[18]
an
d(v) – 2 parallel
s to t, and
=
edges
follows
– 2) > ‘~u~~,(d(v)
which
such
s, while
to
of edges
s to VI and
from
of
all parallel
set
in
achieved
that
v E VI
t with
(d(v)
edges
be
s to v in G’,
connecting
parallel
shown
set VI UVZU{S,
consists
of s, while
lX~/1
edges
edge
v c Vz to
= m’ edges
consider
E.
each
m!
(not
in VI and
graph,
the
two.
where
for
be
+ 1 calls
G’ on vertex
connecting
each
there
m–2n
Lrn – 2nj
out
(5.1)
the
perfect
In what
I can
time
Consider
Mn
I is #P-complete.
computing
of G has degree
counting
been
polynomial
s to
orien-
a bipartite
let
j of the
t.
G’
lL?ll/1~21,
R
Eulerian
from
be
exactly
of
all vertices
that
tions
for
V2, E)
that
of G
where
of s, while
orientations
established
PO
give
Let
such
of
to
is #P-complete.
matchings.
out
and
set
of the
by
since
counting
the
all
Counting
THEOREM 5.1. Exact
tations
Et
denote
necessarily
Now
of Exact
mat thing
connecting
If
R
Hardness
of u. So
are oriented
Lemma
3.3
l~lE112] = 2m, use
(since
and,
V2 that
in Gm,
scheme
2m Hi ri. However, establishing the rapid mixing
property for all these chains remains open.
IPoI
is exactly
out
connecting
Furthermore
. . ..
u E V2 there
orientation
Eulerian
t to V2 are oriented
these
the rest
be oriented
mat thing
connecting
simulating
V2, while
a perfect
v c V1
each perfect
be
lE1/2.
a d~rect
define
can
MC
k <
then
To can be obtained
to obtain
to
s must
all edges
s to
Markov
from
all edges
are balanced,
of n4.
VI
vertex
is oriented
V1 and
G, and clearly
rise to a unique
Let
the
into
graph
each
v to V2 must
u to VI that
between
WINKLER
edges
edge from
connecting
for each vertex
edges
of V2 and
gives
is
lMn,_l]/lMn,l
of n4 simulations
simulating
rate
out
the
original
chain
3.
one out
Hence
scheme
matchings
2@I~je-ni&),
5
– 2 incoming
one incoming
of v. Similarly,
clearly,
all
d(v)
one edge connecting
cient.
G’
t. Consequently
towards
which
AND
(notice
V}.
connection
ations
and
the
be an Eulerian
that
of
G
Realize
is
the
between
number
edges
incident
undirected).
that
of
(undirected)
p
imposes
Let
a
EULERIAN
ORIENTATIONS
OF A GRAPH
decomposition/partition
partition
Let
class
Tk =
that
of
consists
I{p
: p induces
if p induces
exactly
tours
hard
k partition
for
to verify
edges
of
G
{u, w}, pu(e),
one partition
2 Euler
It is not
the
of {e=
145
classes
class,
then
G, so 71 =
where
a
[8] F.
p. (e), . . .}.
} 1. Notice
this
tours)
/2.
Proc.
Math.
Physics
Hence,
’lPol
5.1,
It might
to specify
.
which
some
M.R.
me worth
~k’s
pp
of the
Tk’s
must
investigating
are provably’
6
be hard
[11]
(6) further
[12]
hard.
approximately
count
the complementary
tions
we
for
treat
hardness
complexity
solve
case
Remark
The
capacities
M.R.
of the
on the
results
3 in Section
for
[14]
Extend
edges;
here
(ii)
Classify
the
Euler
tours.
parallelizable?
M.R.
Special
very
thanks
us
to Madhu
due
to
Dominic
discussions,
to
Welsh’s
Sudan,
Prasad
Vijay
Vazirani
for
stages
of this
work.
and
work.
for
to Madhu
Thanks
Tetali,
many
Welsh
[16]
Umesh
discussions
for
also
due
Vazirani,
during
early
Jerrum,
D.
Aldous,
for
On the
Uniform
lated
Markov
CombinatorisJ
Annealing,
[3] A.Z. Broder,
in Eng.
Solved
Academic
Simulation
Distributions
Probability
1987, pp 33-46.
[2] R. Baxter, Exactly
chanics,
chain
Methods
Method
and
and Inf.
SimuSci.
1,
in Statistical
Me-
at Random?
(On
Press, London.
How Hard
the Approximation
is it to Marry
of the Permanent),
STOC
1986, pp
50-58.
[4] P. Dagum
of Graphs
and M. Luby,
with
Large
Approximating
Factors,
the Permanent
Siam
J. on Comp,
to
appear.
[5] P. Dagum,
topes,
FOCS
M. Luby,
M. Mihail,
Permanents
, and
1988, pp 412-421.
[6] P. Diaconis
and D. Strock,
and U. Vazirani,
Graphs
with
Geometric
Large
Bounds
Chains,
preprint.
values of Matkov
[7] M. Dyer, A. Frieze, and R. Kannan,
Polynomial
of Convex
Time
Algorithm
Bodies,STOC
for
Estimating
1989, pp 375-381.
PolyFactors,
for EigenA
Random
Volumes
chains:
resolved,
Fast
ICALP
for
and
the
the
Ap-
STOC
L.G.
1988,
Generation
1989.
Polynomial-time
Ising
Approx-
Model,
Tech.
Report,
V.V.
Vazirani,
(1990).
Valiant,
of
Uniform
Proceedings
the
and
Combinatorial
Theoretical
Distribution,
E.H.
Lieb,
L.
ResiduaJ
162,
Lovisz
Ran-
Structures
from
Computer
Science
a
M.
and
the
[18]
FOCS
Simonovis,
Isoperimetric
1989,
pp
Square
Ice,
Physical
The
1990,
and
Rate
to
and
of
Com-
appear.
Convergence
Treatment
M.R.
Jerrum,
and
of Markov
of
Expanders),
Approximate
rapidly
mixing
and Computing,
Valiant,
Mixing
Inequality,
526-531.
and
generation
The
Theoretical
manent,
FOCS
Combinatorial
Sinclair
L.G.
M.
Conductance
(A
of
An
Volume,
Mihail,
Chains
A.
Entropy
162-171.
Chains,
counting,
markov
chains,
to appear.
Complexity
of Computing
Computer
Science,
the
Map”,
the
Per-
1979, pp 189-
201.
[19]
D. J.A.
Welsh,
[20] D.J.A.
[1]
Systems
of Statistical
Conductance
Markov
A. Sinclair,
Generation
DIMACS
References
and
Information
and
the
M.R.
uniform
some
Sudan
are
for
of Edinburgh,
Markov
[17]
are
enlightening
directing
[15]
4,
Acknowledgements
Sinclair,
A. Sinclair,
Algorithms
puting
Re-
and
Jerrum
Review
(iii)
(iv)
108,
43, 1986, pp 169-188.
ques-
(i)
(1990),
Journal
Permanent
Graphs,
Uniform
as
A.
property
Jerrum
dom
and
as well
following
1.
problems
are our
sample
further:
capacities
of counting
extend
result.
with
with
to
orientations,
to investigate
graphs
the
To what
7
Eulerian
are interesting
the results
schemes
Sot.
the
Poly-
235-243.
imation
[13]
randomized
On
Monomer-Dimer
Intractable,
and
Mixing
of Regular
Summary
efficient
Welsh,
and Tutte
48, pp 121-134.
Jerrum
University
We gave
PM.
2 Dimensional
proximation
k=l
by Theorem
to compute.
= ‘y2’7i
D.J.A.
of Jones
Carob.
are Computationally
that
(y)
and
nomials,
Rapid
H
Vcv
Vertigan,
Complexity
[9] M.R. Jerrum,
[10]
(5.4)
D.L.
35, pp 35-53.
suggests
(#Euler
Jaeger,
Computational
May
Welsh,
“On
The Comput
Classical
Problems
order
Physical
(1990),
in
pp
Tutte
’91 and Dagstuhl
307-321.
from
Systems,
June
ational
Lecture
Notes,
’91.
Complexit
Statistical
Physics,
Oxford
University
y of Some
in
DisPress,