随机算法 (Fall 2011)/Lovász Local Lemma

Lovász Local Lemma

Consider a set of "bad" events [math]\displaystyle{ A_1,A_2,\ldots,A_n }[/math]. Suppose that [math]\displaystyle{ \Pr[A_i]\le p }[/math] for all [math]\displaystyle{ 1\le i\le n }[/math]. We want to show that there is a situation that none of the bad events occurs. Due to the probabilistic method, we need to prove that

[math]\displaystyle{ \Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]\gt 0. }[/math]

Case 1: mutually independent events.

If all the bad events [math]\displaystyle{ A_1,A_2,\ldots,A_n }[/math] are mutually independent, then

[math]\displaystyle{ \Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]\ge(1-p)^n\gt 0, }[/math]

for any [math]\displaystyle{ p\lt 1 }[/math].

Case 2: arbitrarily dependent events.

On the other hand, if we put no assumption on the dependencies between the events, then by the union bound (which holds unconditionally),

[math]\displaystyle{ \Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]=1-\Pr\left[\bigvee_{i=1}^n A_i\right]\ge 1-np, }[/math]

which is not an interesting bound for [math]\displaystyle{ p\ge\frac{1}{n} }[/math]. We cannot improve bound without further information regarding the dependencies between the events.

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We would like to know what is going on between the two extreme cases: mutually independent events, and arbitrarily dependent events. The Lovász local lemma provides such a tool.

The local lemma is powerful tool for showing the possibility of rare event under limited dependencies. The structure of dependencies between a set of events is described by a dependency graph.

Definition

Let [math]\displaystyle{ A_1,A_2,\ldots,A_n }[/math] be a set of events. A graph [math]\displaystyle{ D=(V,E) }[/math] on the set of vertices [math]\displaystyle{ V=\{1,2,\ldots,n\} }[/math] is called a dependency graph for the events [math]\displaystyle{ A_1,\ldots,A_n }[/math] if for each [math]\displaystyle{ i }[/math], [math]\displaystyle{ 1\le i\le n }[/math], the event [math]\displaystyle{ A_i }[/math] is mutually independent of all the events [math]\displaystyle{ \{A_j\mid (i,j)\not\in E\} }[/math].

Example

Let [math]\displaystyle{ X_1,X_2,\ldots,X_m }[/math] be a set of mutually independent random variables. Each event [math]\displaystyle{ A_i }[/math] is a predicate defined on a number of variables among [math]\displaystyle{ X_1,X_2,\ldots,X_m }[/math]. Let [math]\displaystyle{ v(A_i) }[/math] be the unique smallest set of variables which determine [math]\displaystyle{ A_i }[/math]. The dependency graph [math]\displaystyle{ D=(V,E) }[/math] is defined by

[math]\displaystyle{ (i,j)\in E }[/math] iff [math]\displaystyle{ v(A_i)\cap v(A_j)\neq \emptyset }[/math].

The following lemma, known as the Lovász local lemma, first proved by Erdős and Lovász in 1975, is an extremely powerful tool, as it supplies a way for dealing with rare events.

Lovász Local Lemma (symmetric case)

Let [math]\displaystyle{ A_1,A_2,\ldots,A_n }[/math] be a set of events, and assume that the following hold: for all [math]\displaystyle{ 1\le i\le n }[/math], [math]\displaystyle{ \Pr[A_i]\le p }[/math]; the maximum degree of the dependency graph for the events [math]\displaystyle{ A_1,A_2,\ldots,A_n }[/math] is [math]\displaystyle{ d }[/math], and [math]\displaystyle{ ep(d+1)\le 1 }[/math]. Then [math]\displaystyle{ \Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]\gt 0 }[/math].

We will prove a general version of the local lemma, where the events [math]\displaystyle{ A_i }[/math] are not symmetric. This generalization is due to Spencer.

Lovász Local Lemma (general case)

Let [math]\displaystyle{ D=(V,E) }[/math] be the dependency graph of events [math]\displaystyle{ A_1,A_2,\ldots,A_n }[/math]. Suppose there exist real numbers [math]\displaystyle{ x_1,x_2,\ldots, x_n }[/math] such that [math]\displaystyle{ 0\le x_i\lt 1 }[/math] and for all [math]\displaystyle{ 1\le i\le n }[/math], [math]\displaystyle{ \Pr[A_i]\le x_i\prod_{(i,j)\in E}(1-x_j) }[/math]. Then [math]\displaystyle{ \Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]\ge\prod_{i=1}^n(1-x_i) }[/math].

Proof. We can use the following probability identity to compute the probability of the intersection of events: Lemma 1 [math]\displaystyle{ \Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]=\prod_{i=1}^n\Pr\left[\overline{A_i}\mid \bigwedge_{j=1}^{i-1}\overline{A_{j}}\right] }[/math]. Proof. By definition of conditional probability, [math]\displaystyle{ \Pr\left[\overline{A_n}\mid\bigwedge_{i=1}^{n-1}\overline{A_{i}}\right] =\frac{\Pr\left[\bigwedge_{i=1}^n\overline{A_{i}}\right]} {\Pr\left[\bigwedge_{i=1}^{n-1}\overline{A_{i}}\right]} }[/math], so we have [math]\displaystyle{ \Pr\left[\bigwedge_{i=1}^n\overline{A_{i}}\right]=\Pr\left[\bigwedge_{i=1}^{n-1}\overline{A_{i}}\right]\Pr\left[\overline{A_n}\mid\bigwedge_{i=1}^{n-1}\overline{A_{i}}\right] }[/math]. The lemma is proved by recursively applying this equation. [math]\displaystyle{ \square }[/math] Next we prove by induction on [math]\displaystyle{ m }[/math] that for any set of [math]\displaystyle{ m }[/math] events [math]\displaystyle{ i_1,\ldots,i_m }[/math], [math]\displaystyle{ \Pr\left[A_{i_1}\mid \bigwedge_{j=2}^m\overline{A_{i_j}}\right]\le x_{i_1} }[/math]. The local lemma is a direct consequence of this by applying Lemma 1. For [math]\displaystyle{ m=1 }[/math], this is obvious. For general [math]\displaystyle{ m }[/math], let [math]\displaystyle{ i_2,\ldots,i_k }[/math] be the set of vertices adjacent to [math]\displaystyle{ i_1 }[/math] in the dependency graph. Clearly [math]\displaystyle{ k-1\le d }[/math]. And it holds that [math]\displaystyle{ \Pr\left[A_{i_1}\mid \bigwedge_{j=2}^m\overline{A_{i_j}}\right] =\frac{\Pr\left[ A_i\wedge \bigwedge_{j=2}^k\overline{A_{i_j}}\mid \bigwedge_{j=k+1}^m\overline{A_{i_j}}\right]} {\Pr\left[\bigwedge_{j=2}^k\overline{A_{i_j}}\mid \bigwedge_{j=k+1}^m\overline{A_{i_j}}\right]} }[/math], which is due to the basic conditional probability identity [math]\displaystyle{ \Pr[A\mid BC]=\frac{\Pr[AB\mid C]}{\Pr[B\mid C]} }[/math]. We bound the numerator [math]\displaystyle{ \begin{align} \Pr\left[ A_{i_1}\wedge \bigwedge_{j=2}^k\overline{A_{i_j}}\mid \bigwedge_{j=k+1}^m\overline{A_{i_j}}\right] &\le\Pr\left[ A_{i_1}\mid \bigwedge_{j=k+1}^m\overline{A_{i_j}}\right]\\ &=\Pr[A_{i_1}]\\ &\le x_{i_1}\prod_{(i_1,j)\in E}(1-x_j). \end{align} }[/math] The equation is due to the independence between [math]\displaystyle{ A_{i_1} }[/math] and [math]\displaystyle{ A_{i_k+1},\ldots,A_{i_m} }[/math]. The denominator can be expanded using Lemma 1 as [math]\displaystyle{ \Pr\left[\bigwedge_{j=2}^k\overline{A_{i_j}}\mid \bigwedge_{j=k+1}^m\overline{A_{i_j}}\right] =\prod_{j=2}^k\Pr\left[\overline{A_{i_j}}\mid \bigwedge_{\ell=j+1}^m\overline{A_{i_\ell}}\right] }[/math] which by the induction hypothesis, is at least [math]\displaystyle{ \prod_{j=2}^k(1-x_{i_j})=\prod_{\{i_1,i_j\}\in E}(1-x_j) }[/math] where [math]\displaystyle{ E }[/math] is the edge set of the dependency graph. Therefore, [math]\displaystyle{ \Pr\left[A_{i_1}\mid \bigwedge_{j=2}^m\overline{A_{i_j}}\right] \le\frac{x_{i_1}\prod_{(i_1,j)\in E}(1-x_j)}{\prod_{\{i_1,i_j\}\in E}(1-x_j)}\le x_{i_1}. }[/math] Applying Lemma 1, [math]\displaystyle{ \begin{align} \Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right] &=\prod_{i=1}^n\Pr\left[\overline{A_i}\mid \bigwedge_{j=1}^{i-1}\overline{A_{j}}\right]\\ &=\prod_{i=1}^n\left(1-\Pr\left[A_i\mid \bigwedge_{j=1}^{i-1}\overline{A_{j}}\right]\right)\\ &\ge\prod_{i=1}^n\left(1-x_i\right). \end{align} }[/math] [math]\displaystyle{ \square }[/math]

To prove the symmetric case. Let [math]\displaystyle{ x_i=\frac{1}{d+1} }[/math] for all [math]\displaystyle{ i=1,2,\ldots,n }[/math]. Note that [math]\displaystyle{ \left(1-\frac{1}{d+1}\right)^d\gt \frac{1}{\mathrm{e}} }[/math].

If the following conditions are satisfied:

1. for all [math]\displaystyle{ 1\le i\le n }[/math], [math]\displaystyle{ \Pr[A_i]\le p }[/math];
2. [math]\displaystyle{ ep(d+1)\le 1 }[/math];

then for all [math]\displaystyle{ 1\le i\le n }[/math],

[math]\displaystyle{ \Pr[A_i]\le p\le\frac{1}{e(d+1)}\lt \frac{1}{d+1}\left(1-\frac{1}{d+1}\right)^d\le x_i\prod_{(i,j)\in E}(1-x_j) }[/math].

Due to the local lemma for general cases, this implies that

[math]\displaystyle{ \Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]\ge\prod_{i=1}^n(1-x_i)=\left(1-\frac{1}{d+1}\right)^n\gt 0 }[/math].

This gives the symmetric version of local lemma.

Constructive Proof of LLL

We consider a restrictive case.

Let [math]\displaystyle{ X_1,X_2,\ldots,X_m\in\{\mathrm{true},\mathrm{false}\} }[/math] be a set of mutually independent random variables which assume boolean values. Each event [math]\displaystyle{ A_i }[/math] is an AND of at most [math]\displaystyle{ k }[/math] literals ([math]\displaystyle{ X_i }[/math] or [math]\displaystyle{ \neg X_i }[/math]). Let [math]\displaystyle{ v(A_i) }[/math] be the set of the [math]\displaystyle{ k }[/math] variables that [math]\displaystyle{ A_i }[/math] depends on. The probability that none of the bad events occurs is

[math]\displaystyle{ \Pr\left[\bigwedge_{i=1}^n \overline{A_i}\right]. }[/math]

In this particular model, the dependency graph [math]\displaystyle{ D=(V,E) }[/math] is defined as that [math]\displaystyle{ (i,j)\in E }[/math] iff [math]\displaystyle{ v(A_i)\cap v(A_j)\neq \emptyset }[/math].

Observe that [math]\displaystyle{ \overline{A_i} }[/math] is a clause (OR of literals). Thus, [math]\displaystyle{ \bigwedge_{i=1}^n \overline{A_i} }[/math] is a [math]\displaystyle{ k }[/math]-CNF, the CNF that each clause depends on [math]\displaystyle{ k }[/math] variables. The probability

[math]\displaystyle{ \bigwedge_{i=1}^n \overline{A_i}\gt 0 }[/math]

means that the the [math]\displaystyle{ k }[/math]-CNF [math]\displaystyle{ \bigwedge_{i=1}^n \overline{A_i} }[/math] is satisfiable.

The satisfiability of [math]\displaystyle{ k }[/math]-CNF is a hard problem. In particular, 3SAT (the satisfiability of 3-CNF) is the first NP-complete problem (the Cook-Levin theorem). Given the current suspect on NP vs P, we do not expect to solve this problem generally.

However, the condition of the Lovasz local lemma has an extra assumption on the degree of dependency graph. In our model, this means that each clause shares variables with at most [math]\displaystyle{ d }[/math] other clauses. We call a [math]\displaystyle{ k }[/math]-CNF with this property a [math]\displaystyle{ k }[/math]-CNF with bounded degree [math]\displaystyle{ d }[/math].

Therefore, proving the Lovasz local lemma on the restricted forms of events as described above, can be reduced to the following problem:

Problem

Find a condition on [math]\displaystyle{ k }[/math] and [math]\displaystyle{ d }[/math], such that any [math]\displaystyle{ k }[/math]-CNF with bounded degree [math]\displaystyle{ d }[/math] is satisfiable.

In 2009, Moser comes up with the following procedure solving the problem. He later generalizes the procedure to general forms of events. This not only gives a beautiful constructive proof to the Lovasz local lemma, but also provides an efficient randomized algorithm for finding a satisfiable assignment for a number of events with bounded dependencies.

Let [math]\displaystyle{ \phi }[/math] be a [math]\displaystyle{ k }[/math]-CNF of [math]\displaystyle{ n }[/math] clauses with bounded degree [math]\displaystyle{ d }[/math], defined on variables [math]\displaystyle{ X_1,\ldots,X_m }[/math]. The following procedure find a satisfiable assignment for [math]\displaystyle{ \phi }[/math].

Solve([math]\displaystyle{ \phi }[/math])

Pick a random assignment of [math]\displaystyle{ X_1,\ldots,X_m }[/math]. While there is an unsatisfied clause [math]\displaystyle{ C }[/math] in [math]\displaystyle{ \phi }[/math] Fix([math]\displaystyle{ C }[/math]).

The sub-routine Fix is defined as follows:

Fix([math]\displaystyle{ C }[/math])

Replace the variables in [math]\displaystyle{ v(C) }[/math] with new random values. While there is unsatisfied clause [math]\displaystyle{ D }[/math] that [math]\displaystyle{ v(C)\cap v(D)\neq \emptyset }[/math] Fix([math]\displaystyle{ D }[/math]).

The procedure looks very simple. It just recursively fixes the unsatisfied clauses by randomly replacing the assignment to the variables.

We then prove it works.

Number of top-level callings of Fix

In Solve([math]\displaystyle{ \phi }[/math]), the subroutine Fix([math]\displaystyle{ C }[/math]) is called. We now upper bound the number of times it is called (not including the recursive calls).

Assume Fix([math]\displaystyle{ C }[/math]) always terminates.

Observation

Every clause that was satisfied before Fix([math]\displaystyle{ C }[/math]) was called will still remain satisfied and [math]\displaystyle{ C }[/math] will also be satisfied after Fix([math]\displaystyle{ C }[/math]) returns.

The observation can be proved by induction on the structure of recursion. Since there are [math]\displaystyle{ n }[/math] clauses, Solve([math]\displaystyle{ \phi }[/math]) makes at most [math]\displaystyle{ n }[/math] calls to Fix.

We then prove that Fix([math]\displaystyle{ C }[/math]) terminates.

Termination of Fix

The idea of the proof is to reconstruct a random string.

Suppose that during the running of Solve([math]\displaystyle{ \phi }[/math]), the Fix subroutine is called for [math]\displaystyle{ t }[/math] times (including all the recursive calls).

Let [math]\displaystyle{ s }[/math] be the sequence of the random bits used by Solve([math]\displaystyle{ \phi }[/math]). It is easy to see that the length of [math]\displaystyle{ s }[/math] is [math]\displaystyle{ |s|=m+tk }[/math], because the initial random assignment of [math]\displaystyle{ m }[/math] variables takes [math]\displaystyle{ m }[/math] bits, and each time of calling Fix takes [math]\displaystyle{ k }[/math] bits.

We then reconstruct [math]\displaystyle{ s }[/math] in an alternative way.

Recall that Solve([math]\displaystyle{ \phi }[/math]) calls Fix([math]\displaystyle{ C }[/math]) at top-level for at most [math]\displaystyle{ n }[/math] times. Each calling of Fix([math]\displaystyle{ C }[/math]) defines a recursion tree, rooted at clause [math]\displaystyle{ C }[/math], and each node corresponds to a clause (not necessarily distinct, since a clause might be fixed for several times). Therefore, the entire running history of Solve([math]\displaystyle{ \phi }[/math]) can be described by at most [math]\displaystyle{ n }[/math] recursion trees.

Observation 1

Fix a [math]\displaystyle{ \phi }[/math]. The [math]\displaystyle{ n }[/math] recursion trees which capture the total running history of Solve([math]\displaystyle{ \phi }[/math]) can be encoded in [math]\displaystyle{ n\log n+t(\log d+O(1)) }[/math] bits.

Each root node corresponds to a clause. There are [math]\displaystyle{ n }[/math] clauses in [math]\displaystyle{ \phi }[/math]. The [math]\displaystyle{ n }[/math] root nodes can be represented in [math]\displaystyle{ n\log n }[/math] bits.

The smart part is how to encode the branches of the tree. Note that Fix([math]\displaystyle{ C }[/math]) will call Fix([math]\displaystyle{ D }[/math]) only for the [math]\displaystyle{ D }[/math] that shares variables with [math]\displaystyle{ C }[/math]. For a k-CNF with bounded degree [math]\displaystyle{ d }[/math], each clause [math]\displaystyle{ C }[/math] can share variables with at most [math]\displaystyle{ d }[/math] many other clauses. Thus, each branch in the recursion tree can be represented in [math]\displaystyle{ \log d }[/math] bits. There are extra [math]\displaystyle{ O(1) }[/math] bits needed to denote whether the recursion ends. So totally [math]\displaystyle{ n\log n+t(\log d+O(1)) }[/math] bits are sufficient to encode all [math]\displaystyle{ n }[/math] recursion trees.

Observation 2

The random sequence [math]\displaystyle{ s }[/math] can be encoded in [math]\displaystyle{ m+n\log n+t(\log d+O(1)) }[/math] bits.

With [math]\displaystyle{ n\log n+t(\log d+O(1)) }[/math] bits, the structure of all the recursion trees can be encoded. With extra [math]\displaystyle{ m }[/math] bits, the final assignment of the [math]\displaystyle{ m }[/math] variables is stored.

We then observe that with these information, the sequence of the random bits [math]\displaystyle{ s }[/math] can be reconstructed from backwards from the final assignment.

The key step is that a clause [math]\displaystyle{ C }[/math] is only fixed when it is unsatisfied (obvious), and an unsatisfied clause [math]\displaystyle{ C }[/math] must have exact one assignment (a clause is OR of literals, thus has exact one unsatisfied assignment). Thus, each node in the recursion tree tells the [math]\displaystyle{ k }[/math] random bits in the random sequence [math]\displaystyle{ s }[/math] used in the call of the Fix corresponding to the node. Therefore, [math]\displaystyle{ s }[/math] can be reconstructed from the final assignment plus at most [math]\displaystyle{ n }[/math] recursion trees, which can be encoded in at most [math]\displaystyle{ m+n\log n+t(\log d+O(1)) }[/math] bits.

The following theorem lies in the heart of the Kolmogorov complexity. The theorem states that random sequence is incompressible.

Theorem (Kolmogorov)

For any encoding scheme , with high probability, a random sequence [math]\displaystyle{ s }[/math] is encoded in at least [math]\displaystyle{ |s| }[/math] bits.

Applying the theorem, we have that with high probability,

[math]\displaystyle{ m+n\log n+t(\log d+O(1))\ge |s|=m+tk }[/math].

Therefore,

[math]\displaystyle{ t(k-O(1)-\log d)\le n\log n. }[/math]

In order to bound [math]\displaystyle{ t }[/math], we need

[math]\displaystyle{ k-O(1)-\log d\gt 0 }[/math],

which hold for [math]\displaystyle{ d\lt 2^{k-\alpha} }[/math] for some constant [math]\displaystyle{ \alpha\gt 0 }[/math]. In fact, in this case, [math]\displaystyle{ t=O(n\log n) }[/math], the running time of the procedure is bounded by a polynomial!

Back to the local lemma

We showed that for [math]\displaystyle{ d\lt 2^{k-O(1)} }[/math], any [math]\displaystyle{ k }[/math]-CNF with bounded degree [math]\displaystyle{ d }[/math] is satisfiable, and the satisfied assignment can be found within polynomial time with high probability. Now we interprete this in a language of the local lemma.

Recall that the symmetric version of the local lemma:

Theorem (The local lemma: symmetric case)

Let [math]\displaystyle{ A_1,A_2,\ldots,A_n }[/math] be a set of events, and assume that the following hold: for all [math]\displaystyle{ 1\le i\le n }[/math], [math]\displaystyle{ \Pr[A_i]\le p }[/math]; the maximum degree of the dependency graph for the events [math]\displaystyle{ A_1,A_2,\ldots,A_n }[/math] is [math]\displaystyle{ d }[/math], and [math]\displaystyle{ ep(d+1)\le 1 }[/math]. Then [math]\displaystyle{ \Pr\left[\bigwedge_{i=1}^n\overline{A_i}\right]\gt 0 }[/math].

Suppose the underlying probability space is a number of mutually independent uniform random boolean variables, and the evens [math]\displaystyle{ \overline{A_i} }[/math] are clauses defined on [math]\displaystyle{ k }[/math] variables. Then,

[math]\displaystyle{ p=2^{-k} }[/math]

thus, the condition [math]\displaystyle{ ep(d+1)\le 1 }[/math] means that

[math]\displaystyle{ d\lt 2^{k}/e }[/math]

which means the Moser's procedure is asymptotically optimal on the degree of dependency.