Lecture 10 { Algorithmic version of the local lemma

Let A be a collection of random events A1: : : Am. For event Ai, let Γ(Ai) be a minimal set of events that Ai depends on in the sense that Ai is independent of all events in A \ {Γ(Ai) ∪ Ai} (information on events not in Γ(Ai) ∪ Ai does not affect the probability of event Ai happening). The general version of the Lovasz local lemma [4] (see also [1]) is as follows. Theorem 1 If there is an assignment 0 < xi < 1 satisfying for every i: Pr[Ai] ≤ xi jjAj2(Ai) (1 − xj) then the probability that no event Ai happens is at least ∏m i=1(1 − xi)> 0. Observe that if the events were independent, the probability that no bad event happens would have been exactly ∏