📝 Original Info
- Title: Derandomizing the Lovasz Local Lemma more effectively
- ArXiv ID: 0807.2120
- Date: 2008-09-15
- Authors: Researchers from original ArXiv paper
📝 Abstract
The famous Lovasz Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. Kratochvil et al. applied this technique to prove that a k-CNF in which each variable appears at most 2^k/(ek) times is always satisfiable [KST93]. In a breakthrough paper, Beck found that if we lower the occurrences to O(2^(k/48)/k), then a deterministic polynomial-time algorithm can find a satisfying assignment to such an instance [Bec91]. Alon randomized the algorithm and required O(2^(k/8)/k) occurrences [Alo91]. In [Mos06], we exhibited a refinement of his method which copes with O(2^(k/6)/k) of them. The hitherto best known randomized algorithm is due to Srinivasan and is capable of solving O(2^(k/4)/k) occurrence instances [Sri08]. Answering two questions asked by Srinivasan, we shall now present an approach that tolerates O(2^(k/2)/k) occurrences per variable and which can most easily be derandomized. The new algorithm bases on an alternative type of witness tree structure and drops a number of limiting aspects common to all previous methods.
💡 Deep Analysis
Deep Dive into Derandomizing the Lovasz Local Lemma more effectively.
The famous Lovasz Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. Kratochvil et al. applied this technique to prove that a k-CNF in which each variable appears at most 2^k/(ek) times is always satisfiable [KST93]. In a breakthrough paper, Beck found that if we lower the occurrences to O(2^(k/48)/k), then a deterministic polynomial-time algorithm can find a satisfying assignment to such an instance [Bec91]. Alon randomized the algorithm and required O(2^(k/8)/k) occurrences [Alo91]. In [Mos06], we exhibited a refinement of his method which copes with O(2^(k/6)/k) of them. The hitherto best known randomized algorithm is due to Srinivasan and is capable of solving O(2^(k/4)/k) occurrence instances [Sri08]. Answering two questions asked by Srinivasan, we shall now present an approach that tolerates O(2^(k/2)/k) occurrences per variable and which can most easily be derandomized. The new alg
📄 Full Content
arXiv:0807.2120v2 [cs.DS] 15 Sep 2008
Derandomizing
the
Lo
v
ász
Lo
al
Lemma
more
ee tiv
ely
Robin
A.
Moser∗
Institute
for
Theoreti al
Computer
S ien e
Departmen
t
of
Computer
S ien e
ETH
Züri
h,
8092
Züri
h,
Switzerland
robin.moser inf.ethz. h
July
2008
Abstra t
The
famous
Lo
v
ász
Lo
al
Lemma
[EL75℄
is
a
p
o
w
erful
to
ol
to
non- onstru tiv
ely
pro
v
e
the
existen e
of
om
binatorial
ob
je ts
meeting
a
pres rib
ed
olle tion
of
riteria.
Krato
h
víl
et
al.
applied
this
te
hnique
to
pro
v
e
that
a k
-CNF
in
whi
h
ea
h
v
ariable
app
ears
at
most 2k/(ek)
times
is
alw
a
ys
satis able
[KST93
℄.
In
a
breakthrough
pap
er,
Be
k
found
that
if
w
e
lo
w
er
the
o
urren es
to O(2k/48/k),
then
a
deterministi
p
olynomial-time
algorithm
an
nd
a
satisfying
assignmen
t
to
su
h
an
instan e
[Be 91
℄.
Alon
randomized
the
algorithm
and
required O(2k/8/k)
o
urren es
[Alo91
℄.
In
[Mos06
℄,
w
e
exhibited
a
re nemen
t
of
his
metho
d
whi
h
op
es
with O(2k/6/k)
of
them.
The
hitherto
b
est
kno
wn
randomized
algorithm
is
due
to
Sriniv
asan
and
is
apable
of
solving O(2k/4/k)
o
urren e
instan es
[Sri08
℄.
Answ
ering
t
w
o
questions
ask
ed
b
y
Sriniv
asan,
w
e
shall
no
w
presen
t
an
approa
h
that
tolerates O(2k/2/k)
o
urren es
p
er
v
ariable
and
whi
h
an
most
easily
b
e
derandomized.
The
new
algorithm
bases
on
an
alternativ
e
t
yp
e
of
witness
tree
stru ture
and
drops
a
n
um
b
er
of
limiting
asp
e ts
ommon
to
all
previous
metho
ds.
Key
W
ords
and
Phrases.
Lo
v
ász
Lo
al
Lemma,
derandomization,
b
ounded
o
urren e
SA
T
instan es,
h
yp
ergraph
olouring.
1
In
tro
du tion
W
e
assume
an
in nite
supply
of
prop
ositional
variables.
A
liter
al L
is
a
v
ariable x
or
a
omplemen
ted
v
ariable ¯x.
A
nite
set D
of
literals
o
v
er
pairwise
distin t
v
ariables
is
alled
a
lause.
W
e
sa
y
that
a
v
ariable x
o
urs
in D
if x ∈D
or ¯x ∈D
.
A
nite
set ϕ
of
lauses
is
alled
a
formula
or
a
CNF
(Conjun tiv
e
Normal
F
orm).
W
e
sa
y
that ϕ
is
a k
-CNF,
if
ev
ery
lause
has
size
exa tly k
.
W
e
sa
y
that
the
v
ariable x
o
urs j
times
in
a
form
ula
if
there
are
exa tly j
lauses
in
whi
h x
o
urs.
W
e
write
v
ar(ϕ)
to
denote
the
set
of
all
v
ariables
o
urring
in ϕ.
A
truth
assignment
is
a
fun tion α :
v
ar(ϕ) →{0, 1}
whi
h
assigns
a
b
o
olean
v
alue
to
ea
h
v
ariable.
A
literal L = x
(or L = ¯x)
is
satis e
d
b
y α
if α(x) = 1
(or α(x) = 0).
A
lause
is
satis e
d
b
y α
if
it
on
tains
a
satis ed
literal
and
a
form
ula
is
satis e
d
if
all
of
its
lauses
are.
A
form
ula
is
satis able
if
there
exists
a
satisfying
truth
assignmen
t
to
its
v
ariables.
In
[KST93
℄,
Krato
h
víl
et
al.
ha
v
e
applied
the
Lo
v
ász
Lo
al
Lemma
(from
[EL75
℄)
to
pro
v
e
that
ev
ery
k
-CNF
in
whi
h
ev
ery
v
ariable
o
urs
no
more
than 2k/(ek)
times
has
a
satisfying
assignmen
t.
The
Lo
al
∗
Resear
h
is
supp
orted
b
y
the
SNF
Gran
t
200021-118001/1
1
Lemma
urren
tly
app
ears
to
b
e
the
only
w
ork
able
to
ol
for
the
obten
tion
of
su
h
a
b
ound.
Unfortunately
,
all
kno
wn
pro
ofs
based
on
the
Lo
al
Lemma
are
non- onstru tiv
e
and
do
not
dire tly
allo
w
for
the
onstru tion
of
an
e ien
t
algorithm
to
a tually
nd
a
satisfying
assignmen
t.
Su
h
an
algorithm
w
as
rst
pro
vided
b
y
Be
k
in
[Be 91
℄
and
then
randomized
b
y
Alon
[Alo91℄.
Their
algorithm
bases
on
the
prin iple
of
sele ting
a
preliminary
assignmen
t
and
then
dis riminating
lauses
a ording
to
whether
few
or
man
y
of
their
literals
are
satis ed
b
y
it.
The
b
ad
lauses
on
taining
few
satis ed
literals
are
then
reassessed
and
their
v
ariables
are
reassigned
new
v
alues.
Su
h
a
pro
edure
is
e ien
t
if
the
dep
enden ies
are
lo
w
enough
so
as
to
guaran
tee
that
lustered
omp
onen
ts
of
bad
lauses
are
v
ery
small
and
an
b
e
solv
ed
in
a
brute
for e
fashion.
The
original
approa
h
b
y
Be
k
required
that
no
v
ariable
app
ear
a
n
um
b
er
higher
than O(2k/48/k)
of
times.
In
Alon’s
simpli ation,
the
requiremen
t
w
as
still O(2k/8/k).
Sev
eral
authors
ha
v
e
impro
v
ed
and
extended
those
approa
hes,
with
goals
somewhat
omplemen
tary
to
what
w
e
are
on erned
with.
In
[MR98℄
e.g.,
Mollo
y
and
Reed
extrap
olate
guidelines
whi
h
allo
w
for
the
onstru tion
of
an
algorithmi
v
ersion
to
n
umerous
appli ations
of
the
Lo
al
Lemma
other
than
b
ounded
o
urren e
SA
T
or
h
yp
ergraph
2- olouring
(b
eing
almost
iden
ti al
problems).
Czuma
j
and
S
heideler
giv
e
alternativ
e
approa
hes
that
allo
w
for
non-uniform
lause-
or
edge-sizes
[CS00
℄.
W
e
do
not
in
v
estigate
these
v
ariations
more
losely
as
our
urren
t
goal
is
to
further
impro
v
e
on
the
o
urren e
b
ound.
The
hitherto
most
p
o
w
erful
approa
h
has
b
een
re en
tly
published
b
y
Sriniv
asan
[Sri08
℄.
It
allo
ws
to
nd
a
satisfying
assignmen
t
to
a k
-CNF
in
whi
h
ev
ery
v
ariable
oun
ts
no
more
than O(2k/4/k)
o
urren es.
Sriniv
asan
a
hiev
es
the
impro
v
emen
t
b
y
riti ally
augmen
ting
and
re ning
the 2, 3-tree
based
witness
stru tures
used
b
y
Alon.
The
resulting
algorithm
is
inheren
tly
randomized.
Our
o
…(Full text truncated)…
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