Lecture 11: Generalized Lovász Local Lemma. Lovász Local Lemma

Transcription

1 Lecture 11: Generalized

2 Recall We design an experiment with independent random variables X 1,..., X m We define bad events A 1,..., A n where) the bad event A i depends on the variables (X k1,..., X kni We define Vbl i = {k 1,..., k ni }, the set of all variables that the bad event A i depends on The bad event A i can depend on the bad event A j if Vbl i Vbl j Suppose each bad event A i depends on at most d other bad events Suppose we show that, for each bad event A i, the probability of its occurrence P [A i ] p If ep(d + 1) 1, then ( P [ A 1,... A n ] 1 1 ) n > 0 d + 1

3 Generalized We design an experiment with independent random variables X 1,..., X m We define bad events A 1,..., A n Let D i be the set of indices of bad events that A i depends on Suppose we exhibit the existence of numbers (x 1,..., x n ) such that the following holds. For each i, we have: P [A i ] x i j D i (1 x j ) Then the following holds. P [ A 1,..., A n ] n (1 x i ) > 0 i=1

4 Notes Prove using Generalized Lovász Local Lemma The numbers (x 1,..., x n ) are not probabilities that add up to 1. This is an incorrect intuition Prove the following corollary of the generalized Lovász Local Lemma Corollary If, for all i, we have j D i P [ A j ] < 1/4, then P [ A 1,..., A n ] n (1 2P [A i ]) > 0 i=1 Prove the results in the previous lecture using this corollary albeit with slightly worse parameters

5 Frugal Coloring I Definition (Frugal Coloring) A β-frugal Coloring of a graph satisfies the following two conditions 1 It is a valid coloring, and 2 In the neighborhood N(v) of any vertex v, there are at most β vertices with the same color. For example, a 1-frugal coloring of G is a coloring of G 2

6 Frugal Coloring II We will show the following result Theorem For β N, and a graph G with maximum degree β β there exists a β-frugal coloring using /β colors. Note that a graph with maximum degree can be 1-frugally colored with colors. We will prove the general result using

7 Frugal Coloring III Randomly color the vertices of the graph using C colors. We will consider two types of bad events. A e, where e E(G): If the two vertices at the endpoints of the edge e receive the same color. These will be called type-1 bad events. B {u1,...,u β+1 }, where u 1,..., u β+1 V (G): Suppose there exists a vertex v such that u 1,..., u β+1 are distinct vertices in N(G) with identical color. These will be called type-2 bad events.

8 Frugal Coloring IV Note that one type-1 bad event A e can depends on at most 2 other type-1 bad events A e We are now interested in computing how many type-2 bad events can A e depend on. Consider a type-2 bad event B u1,...,u β+1 such that that are in u 1,..., u β+1 N(v). Suppose e = (a, b). Note that a has at most neighbors. So, there ( are at ) most possible ways of choosing v. Note that we have ways of choosing the remaining vertices β {u 1,..., u β+1 ( } \ ) {a}. Similar case for b as well. So, there are at most 2 type-2 events that A β e can depends on.

9 Frugal Coloring V Similarly, a type-2 event B u1,...,u β+1 can depends on (β + 1) ( ) other type-1 bad events and (β + 1) β other type-2 bad events Note that P [A e ] 1 C ] P [B u1,...,u β+1 1 C β So, to prove that a β-frugal coloring exists, it suffices to prove that ( ) (β + 1) (β + 1) C β C β < 1 4

10 Frugal Coloring VI ( ) n We can use the upper bound k the expression (β + 1) C + ( ) en k k to upper-bound ( ) (β + 1) C β β This is left as an exercise

11 Moser-Tardos Algorithm Now we are interested in computing the solution that is guaranteed by. function Seq_LLL(X = {X 1,..., X m }, A = {A 1,..., A n }) X Random Evaluation while i s.t. A i is satisfied do Pick arbitrary A i that is satisfied Re-sample all X j such that j Vbl i end while Output X end function

12 Performance of Moser-Tardos Theorem Suppose there exists (x 1,..., x n ) such that, for all i [n], we have P [A i ] x i j D i (1 x j ). Then the expected number of times sequential Moser-Tardos samples the event A i is at most x i /(1 x i ) and, hence, the expected number of execution of the inner loop is at most i [n] x i/(1 x i ).

13 Parallel Moser-Tardos Algorithm function Parallel_LLL(X, A) X Random Evaluation while i s.t. A i is satisfied do Let S be a maximal independent set in the dependency graph restricted to satisfied A i s X k S Vbl k end while Output X end function Random Evaluation

14 Performance of Parallel Moder-Tardos Theorem Suppose there exists an ε > 0 and (x 1,..., x n ) such that P [A i ] (1 ε)x i j D i (1 x j ). The expected number of inner loops before all bad events are avoided is at most O ( 1 ε i [n] x i 1 x i ).