Outline
The Lovasz Local Lemma
Haritha Eruvuru1
1 Lane Department of Computer Science and Electrical Engineering
West Virginia University
24 April, 2012
Haritha Eruvuru
Randomized Algorithms
Outline
Outline
1
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Outline
1
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Introduction
Introduction
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Introduction
Introduction
Let E1 , E2 . . . En be a set of bad events in probability space.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Introduction
Introduction
Let E1 , E2 . . . En be a set of bad events in probability space.
We need to prove that there is an element in the sample space that is not included in
any of the bad events.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Introduction
Introduction
Let E1 , E2 . . . En be a set of bad events in probability space.
We need to prove that there is an element in the sample space that is not included in
any of the bad events.
Let the events be mutually independent. Hence,
Haritha Eruvuru
Randomized Algorithms
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Introduction
Introduction
Let E1 , E2 . . . En be a set of bad events in probability space.
We need to prove that there is an element in the sample space that is not included in
any of the bad events.
Let the events be mutually independent. Hence,
Y
P(∩i∈I Ei ) =
P(Ei )
i∈I
where I ⊂ [1, n]
Haritha Eruvuru
Randomized Algorithms
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Introduction
Introduction
Let E1 , E2 . . . En be a set of bad events in probability space.
We need to prove that there is an element in the sample space that is not included in
any of the bad events.
Let the events be mutually independent. Hence,
Y
P(∩i∈I Ei ) =
P(Ei )
i∈I
where I ⊂ [1, n]
Also if P(Ei ) < 1 for all i, then
Haritha Eruvuru
Randomized Algorithms
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Introduction
Introduction
Let E1 , E2 . . . En be a set of bad events in probability space.
We need to prove that there is an element in the sample space that is not included in
any of the bad events.
Let the events be mutually independent. Hence,
Y
P(∩i∈I Ei ) =
P(Ei )
i∈I
where I ⊂ [1, n]
Also if P(Ei ) < 1 for all i, then
P(∩ni=1 E i ) =
n
Y
P(E i ) > 0
i=1
Haritha Eruvuru
Randomized Algorithms
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Introduction
Introduction
Let E1 , E2 . . . En be a set of bad events in probability space.
We need to prove that there is an element in the sample space that is not included in
any of the bad events.
Let the events be mutually independent. Hence,
Y
P(∩i∈I Ei ) =
P(Ei )
i∈I
where I ⊂ [1, n]
Also if P(Ei ) < 1 for all i, then
P(∩ni=1 E i ) =
n
Y
P(E i ) > 0
i=1
There exists an element of the sample space that is not included in any bad event.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Introduction
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Introduction
The Lovasz local lemma generalizes the argument to the case where n events are not
mutually independent but the dependency is limited.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Introduction
The Lovasz local lemma generalizes the argument to the case where n events are not
mutually independent but the dependency is limited.
An event E is mutually independent of the events E1 , E2 , . . . En if for any subset I ⊂
[1,n],
P(E | ∩j∈I Ej ) = P(E)
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Introduction
The Lovasz local lemma generalizes the argument to the case where n events are not
mutually independent but the dependency is limited.
An event E is mutually independent of the events E1 , E2 , . . . En if for any subset I ⊂
[1,n],
P(E | ∩j∈I Ej ) = P(E)
The dependency between events can be represented in terms of a dependency graph.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Definition
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Definition
A dependency graph for a set of events E1 , E2 , . . . En is a graph G = (V , E) such that
V = {1 . . . n} and for i = 1 . . . n, event Ei is mutually independent of the events
{Ej | (i, j) ∈
/ E}
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Theorem
Let E1 . . . En be a set of events and assume the following hold:
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Theorem
Let E1 . . . En be a set of events and assume the following hold:
1. for all i, P(Ei ) ≤ p
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Theorem
Let E1 . . . En be a set of events and assume the following hold:
1. for all i, P(Ei ) ≤ p
2. the degree of the dependency graph given by E1 . . . , En is bounded by d
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Theorem
Let E1 . . . En be a set of events and assume the following hold:
1. for all i, P(Ei ) ≤ p
2. the degree of the dependency graph given by E1 . . . , En is bounded by d
3. 4 · d · p ≤ 1
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Theorem
Let E1 . . . En be a set of events and assume the following hold:
1. for all i, P(Ei ) ≤ p
2. the degree of the dependency graph given by E1 . . . , En is bounded by d
3. 4 · d · p ≤ 1
Then,
P(∩ni=1 E i ) ≥ 0
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Theorem
Let E1 . . . En be a set of events and assume the following hold:
1. for all i, P(Ei ) ≤ p
2. the degree of the dependency graph given by E1 . . . , En is bounded by d
3. 4 · d · p ≤ 1
Then,
P(∩ni=1 E i ) ≥ 0
Proof
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Theorem
Let E1 . . . En be a set of events and assume the following hold:
1. for all i, P(Ei ) ≤ p
2. the degree of the dependency graph given by E1 . . . , En is bounded by d
3. 4 · d · p ≤ 1
Then,
P(∩ni=1 E i ) ≥ 0
Proof
Assume S ⊂ {1 . . . n}
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Theorem
Let E1 . . . En be a set of events and assume the following hold:
1. for all i, P(Ei ) ≤ p
2. the degree of the dependency graph given by E1 . . . , En is bounded by d
3. 4 · d · p ≤ 1
Then,
P(∩ni=1 E i ) ≥ 0
Proof
Assume S ⊂ {1 . . . n}
By induction on s = 0, . . . n − 1 we prove that if |S | ≤ s , then for all k ∈
/S,
P(Ek | ∩j∈S E j ) ≤ 2 · p
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Theorem
Let E1 . . . En be a set of events and assume the following hold:
1. for all i, P(Ei ) ≤ p
2. the degree of the dependency graph given by E1 . . . , En is bounded by d
3. 4 · d · p ≤ 1
Then,
P(∩ni=1 E i ) ≥ 0
Proof
Assume S ⊂ {1 . . . n}
By induction on s = 0, . . . n − 1 we prove that if |S | ≤ s , then for all k ∈
/S,
P(Ek | ∩j∈S E j ) ≤ 2 · p
Also, for this to be well defined when S is not empty
P(Ek | ∩j∈S E j ) ≥ 0
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
True for s = 1 as P(E j ) ≥ 1 − p ≥ 0
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
True for s = 1 as P(E j ) ≥ 1 − p ≥ 0
For s ≥ 1, that is S = {1, 2, . . . s}, then
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
True for s = 1 as P(E j ) ≥ 1 − p ≥ 0
For s ≥ 1, that is S = {1, 2, . . . s}, then
s
Y
P ∩si=1 E i =
P E i | ∩i−1
Ej
j=1
i=1
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
True for s = 1 as P(E j ) ≥ 1 − p ≥ 0
For s ≥ 1, that is S = {1, 2, . . . s}, then
s
Y
P ∩si=1 E i =
P E i | ∩i−1
Ej
j=1
i=1
=
s
Y
(1 − P Ei | ∩i−1
Ej )
j=1
i=1
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
True for s = 1 as P(E j ) ≥ 1 − p ≥ 0
For s ≥ 1, that is S = {1, 2, . . . s}, then
s
Y
P ∩si=1 E i =
P E i | ∩i−1
Ej
j=1
i=1
=
s
Y
(1 − P Ei | ∩i−1
Ej )
j=1
i=1
Using induction hypothesis,
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
True for s = 1 as P(E j ) ≥ 1 − p ≥ 0
For s ≥ 1, that is S = {1, 2, . . . s}, then
s
Y
P ∩si=1 E i =
P E i | ∩i−1
Ej
j=1
i=1
=
s
Y
(1 − P Ei | ∩i−1
Ej )
j=1
i=1
Using induction hypothesis,
≥
s
Y
(1 − 2 · p) ≥ 0
i=1
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Let S1 = {j ∈ S |(k, j) ∈ E} and S2 = S − S1 .
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Let S1 = {j ∈ S |(k, j) ∈ E} and S2 = S − S1 .
If S2 = S, then Ek is mutually independent of the events E i , i ∈ S
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Let S1 = {j ∈ S |(k, j) ∈ E} and S2 = S − S1 .
If S2 = S, then Ek is mutually independent of the events E i , i ∈ S
P Ek | ∩j∈S E j = P(Ek ) ≤ p
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Let S1 = {j ∈ S |(k, j) ∈ E} and S2 = S − S1 .
If S2 = S, then Ek is mutually independent of the events E i , i ∈ S
P Ek | ∩j∈S E j = P(Ek ) ≤ p
Let |S2 | ≤ s
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Let S1 = {j ∈ S |(k, j) ∈ E} and S2 = S − S1 .
If S2 = S, then Ek is mutually independent of the events E i , i ∈ S
P Ek | ∩j∈S E j = P(Ek ) ≤ p
Let |S2 | ≤ s
Let Fs be Fs = ∩j∈S E j
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Let S1 = {j ∈ S |(k, j) ∈ E} and S2 = S − S1 .
If S2 = S, then Ek is mutually independent of the events E i , i ∈ S
P Ek | ∩j∈S E j = P(Ek ) ≤ p
Let |S2 | ≤ s
Let Fs be Fs = ∩j∈S E j
Also Fs = Fs1 ∩ Fs2
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Let S1 = {j ∈ S |(k, j) ∈ E} and S2 = S − S1 .
If S2 = S, then Ek is mutually independent of the events E i , i ∈ S
P Ek | ∩j∈S E j = P(Ek ) ≤ p
Let |S2 | ≤ s
Let Fs be Fs = ∩j∈S E j
Also Fs = Fs1 ∩ Fs2
Applying Conditional Probability,
P(Ek | Fs ) =
Haritha Eruvuru
P(Ek ∩ Fs )
P(Fs )
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
P(Ek ∩ Fs ) = P(Ek ∩ Fs1 ∩ Fs2 )
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
P(Ek ∩ Fs ) = P(Ek ∩ Fs1 ∩ Fs2 )
P(Ek ∩ Fs1 | Fs2 ) · P(Fs2 )
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
P(Ek ∩ Fs ) = P(Ek ∩ Fs1 ∩ Fs2 )
P(Ek ∩ Fs1 | Fs2 ) · P(Fs2 )
P(Fs ) = P(Fs1 ∩ Fs2 )
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
P(Ek ∩ Fs ) = P(Ek ∩ Fs1 ∩ Fs2 )
P(Ek ∩ Fs1 | Fs2 ) · P(Fs2 )
P(Fs ) = P(Fs1 ∩ Fs2 )
= P(Fs1 | Fs2 ) · P(Fs2 )
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
P(Ek ∩ Fs ) = P(Ek ∩ Fs1 ∩ Fs2 )
P(Ek ∩ Fs1 | Fs2 ) · P(Fs2 )
P(Fs ) = P(Fs1 ∩ Fs2 )
= P(Fs1 | Fs2 ) · P(Fs2 )
Hence,
P(Ek | Fs ) =
P(Ek ∩ Fs1 | Fs2 )
P(Fs1 |Fs2 )
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
The probability of an intersection of events is bounded by the probability of any one of
the events and as Ek is independent of the events in S2 ,
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
The probability of an intersection of events is bounded by the probability of any one of
the events and as Ek is independent of the events in S2 ,
P(Ek ∩ Fs1 |Fs2 ) ≤ P(Ek |Fs2 ) = P(Ek ) ≤ p
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
The probability of an intersection of events is bounded by the probability of any one of
the events and as Ek is independent of the events in S2 ,
P(Ek ∩ Fs1 |Fs2 ) ≤ P(Ek |Fs2 ) = P(Ek ) ≤ p
|S2 | ≤ |S | = s, applying induction hypothesis to,
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
The probability of an intersection of events is bounded by the probability of any one of
the events and as Ek is independent of the events in S2 ,
P(Ek ∩ Fs1 |Fs2 ) ≤ P(Ek |Fs2 ) = P(Ek ) ≤ p
|S2 | ≤ |S | = s, applying induction hypothesis to,
P(Ei |Fs2 ) = P(Ei | ∩j∈S2 E j )
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Establishing a lower bound on the denominator using |S1 | ≤ d
P(Fs1 |Fs2 ) = P(∩i∈s1 E i | ∩j∈s2 E j )
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Establishing a lower bound on the denominator using |S1 | ≤ d
P(Fs1 |Fs2 ) = P(∩i∈s1 E i | ∩j∈s2 E j )
≥1−
X
P(Ei | ∩j∈s2 E j )
i∈s1
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Establishing a lower bound on the denominator using |S1 | ≤ d
P(Fs1 |Fs2 ) = P(∩i∈s1 E i | ∩j∈s2 E j )
≥1−
X
P(Ei | ∩j∈s2 E j )
i∈s1
≥1−
X
2·p
i∈s1
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Establishing a lower bound on the denominator using |S1 | ≤ d
P(Fs1 |Fs2 ) = P(∩i∈s1 E i | ∩j∈s2 E j )
≥1−
X
P(Ei | ∩j∈s2 E j )
i∈s1
≥1−
X
2·p
i∈s1
≥1−2·p·d
Haritha Eruvuru
Randomized Algorithms
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Lovasz Local Lemma
Proof(Cont.)
Establishing a lower bound on the denominator using |S1 | ≤ d
P(Fs1 |Fs2 ) = P(∩i∈s1 E i | ∩j∈s2 E j )
≥1−
X
P(Ei | ∩j∈s2 E j )
i∈s1
≥1−
X
2·p
i∈s1
≥1−2·p·d
≥
Haritha Eruvuru
1
2
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Using the upper bound and lower bound,
P(Ek |Fs ) =
P(Ek ∩ Fs1 |Fs2 )
P(Fs1 |Fs2 )
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Using the upper bound and lower bound,
P(Ek |Fs ) =
≤
P(Ek ∩ Fs1 |Fs2 )
P(Fs1 |Fs2 )
p
=2·p
1/2
Hence,
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Using the upper bound and lower bound,
P(Ek |Fs ) =
≤
P(Ek ∩ Fs1 |Fs2 )
P(Fs1 |Fs2 )
p
=2·p
1/2
Hence,
P(∩ni=1 E i ) =
n
Y
Ej )
P(E i | ∩i−1
j=1
i=1
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Using the upper bound and lower bound,
P(Ek |Fs ) =
≤
P(Ek ∩ Fs1 |Fs2 )
P(Fs1 |Fs2 )
p
=2·p
1/2
Hence,
P(∩ni=1 E i ) =
n
Y
Ej )
P(E i | ∩i−1
j=1
i=1
=
n
Y
(1 − P(Ei | ∩i−1
E j ))
j=1
i=1
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Lovasz Local Lemma
Proof(Cont.)
≥
n
Y
(1 − 2 · p) ≥ 0
i=1
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Outline
1
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Edge Disjoint Paths
Note
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Edge Disjoint Paths
Note
Let there exist n pairs of users need to communicate using edge-disjoint paths on a
network.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Edge Disjoint Paths
Note
Let there exist n pairs of users need to communicate using edge-disjoint paths on a
network.
Each pair i = 1 . . . n can choose path from a collection Fi from m paths
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Edge Disjoint Paths
Note
Let there exist n pairs of users need to communicate using edge-disjoint paths on a
network.
Each pair i = 1 . . . n can choose path from a collection Fi from m paths
Theorem
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Edge Disjoint Paths
Note
Let there exist n pairs of users need to communicate using edge-disjoint paths on a
network.
Each pair i = 1 . . . n can choose path from a collection Fi from m paths
Theorem
If any path in Fi shares edges with no more than k paths in Fj , where i 6= j and
8·n·k
≤ 1 , then there is a way to choose n edge-disjoint paths connecting the n pairs.
m
Proof
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Edge Disjoint Paths
Note
Let there exist n pairs of users need to communicate using edge-disjoint paths on a
network.
Each pair i = 1 . . . n can choose path from a collection Fi from m paths
Theorem
If any path in Fi shares edges with no more than k paths in Fj , where i 6= j and
8·n·k
≤ 1 , then there is a way to choose n edge-disjoint paths connecting the n pairs.
m
Proof
Let the probability space be defined by each pair choosing a path independently and
uniformly at random from its set of m paths.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Edge Disjoint Paths
Note
Let there exist n pairs of users need to communicate using edge-disjoint paths on a
network.
Each pair i = 1 . . . n can choose path from a collection Fi from m paths
Theorem
If any path in Fi shares edges with no more than k paths in Fj , where i 6= j and
8·n·k
≤ 1 , then there is a way to choose n edge-disjoint paths connecting the n pairs.
m
Proof
Let the probability space be defined by each pair choosing a path independently and
uniformly at random from its set of m paths.
Let Ei,j be the event that the paths chosen by i and j share at least one edge.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Edge Disjoint Paths
Note
Let there exist n pairs of users need to communicate using edge-disjoint paths on a
network.
Each pair i = 1 . . . n can choose path from a collection Fi from m paths
Theorem
If any path in Fi shares edges with no more than k paths in Fj , where i 6= j and
8·n·k
≤ 1 , then there is a way to choose n edge-disjoint paths connecting the n pairs.
m
Proof
Let the probability space be defined by each pair choosing a path independently and
uniformly at random from its set of m paths.
Let Ei,j be the event that the paths chosen by i and j share at least one edge.
Path in Fi shares edges with no more than k paths in Fj . Hence,
p = P(Ei,j ) ≤
Haritha Eruvuru
k
m
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Outline
1
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Edge Disjoint Paths
Proof(Cont.)
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Edge Disjoint Paths
Proof(Cont.)
Let d be the degree of the dependency graph.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Edge Disjoint Paths
Proof(Cont.)
Let d be the degree of the dependency graph.
0
0
Since event Ei,j is independent of all events Ei 0 ,j 0 when i ∈
/ i, j and j ∈
/ i, j, then
d ≤ 2 · n.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Edge Disjoint Paths
Proof(Cont.)
Let d be the degree of the dependency graph.
0
0
Since event Ei,j is independent of all events Ei 0 ,j 0 when i ∈
/ i, j and j ∈
/ i, j, then
d ≤ 2 · n.
As
4·d ·p ≤
8·n·k
≤1
m
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Edge Disjoint Paths
Proof(Cont.)
Let d be the degree of the dependency graph.
0
0
Since event Ei,j is independent of all events Ei 0 ,j 0 when i ∈
/ i, j and j ∈
/ i, j, then
d ≤ 2 · n.
As
4·d ·p ≤
8·n·k
≤1
m
All the conditions of Lovasz local lemma are satisfied and hence
P(πi ∈j
/ E i,j ) ≥ 0
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Edge Disjoint Paths
Proof(Cont.)
Let d be the degree of the dependency graph.
0
0
Since event Ei,j is independent of all events Ei 0 ,j 0 when i ∈
/ i, j and j ∈
/ i, j, then
d ≤ 2 · n.
As
4·d ·p ≤
8·n·k
≤1
m
All the conditions of Lovasz local lemma are satisfied and hence
P(πi ∈j
/ E i,j ) ≥ 0
Hence there is a choice of paths such that the n paths are edge disjoint.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Outline
1
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Satisfiability
Theorem
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Satisfiability
Theorem
If no variable in a k-SAT formula appears in more than T = 2k /4 · k clauses, then the
formula has a satisfying assignment.
Proof
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Satisfiability
Theorem
If no variable in a k-SAT formula appears in more than T = 2k /4 · k clauses, then the
formula has a satisfying assignment.
Proof
Let there be a probability space defined by some variables that are randomly assigned.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Satisfiability
Theorem
If no variable in a k-SAT formula appears in more than T = 2k /4 · k clauses, then the
formula has a satisfying assignment.
Proof
Let there be a probability space defined by some variables that are randomly assigned.
Let Ei for i = 1 . . . m be an event that the i th clause is not satisfied by the random
assignment.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Satisfiability
Theorem
If no variable in a k-SAT formula appears in more than T = 2k /4 · k clauses, then the
formula has a satisfying assignment.
Proof
Let there be a probability space defined by some variables that are randomly assigned.
Let Ei for i = 1 . . . m be an event that the i th clause is not satisfied by the random
assignment.
As each clause has k literals,
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Satisfiability
Theorem
If no variable in a k-SAT formula appears in more than T = 2k /4 · k clauses, then the
formula has a satisfying assignment.
Proof
Let there be a probability space defined by some variables that are randomly assigned.
Let Ei for i = 1 . . . m be an event that the i th clause is not satisfied by the random
assignment.
As each clause has k literals,
P(Ei ) = 2−k
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Satisfiability
Theorem
If no variable in a k-SAT formula appears in more than T = 2k /4 · k clauses, then the
formula has a satisfying assignment.
Proof
Let there be a probability space defined by some variables that are randomly assigned.
Let Ei for i = 1 . . . m be an event that the i th clause is not satisfied by the random
assignment.
As each clause has k literals,
P(Ei ) = 2−k
Ei is independent of all the events related to clauses that do not share variables having
clause i.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Satisfiability
Theorem
If no variable in a k-SAT formula appears in more than T = 2k /4 · k clauses, then the
formula has a satisfying assignment.
Proof
Let there be a probability space defined by some variables that are randomly assigned.
Let Ei for i = 1 . . . m be an event that the i th clause is not satisfied by the random
assignment.
As each clause has k literals,
P(Ei ) = 2−k
Ei is independent of all the events related to clauses that do not share variables having
clause i.
Each variable in clause i cannot appear in more than T = 2k /4 · k clauses.
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Satisfiability
Theorem
If no variable in a k-SAT formula appears in more than T = 2k /4 · k clauses, then the
formula has a satisfying assignment.
Proof
Let there be a probability space defined by some variables that are randomly assigned.
Let Ei for i = 1 . . . m be an event that the i th clause is not satisfied by the random
assignment.
As each clause has k literals,
P(Ei ) = 2−k
Ei is independent of all the events related to clauses that do not share variables having
clause i.
Each variable in clause i cannot appear in more than T = 2k /4 · k clauses.
Hence the degree of dependency graph is bounded by d ≤ k · T ≤ 2k−2 .
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Satisfiability
Theorem(Cont.)
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Satisfiability
Theorem(Cont.)
Applying Lovasz local lemma,
Haritha Eruvuru
Randomized Algorithms
Lovasz Local Lemma
Introduction
Edge Disjoint Paths
Edge Disjoint Paths
Satisfiability
Satisfiability
Theorem(Cont.)
Applying Lovasz local lemma,
P(∩m
i=1 E i ) ≥ 0
Haritha Eruvuru
Randomized Algorithms