An interactive version of the Lov\'asz local lemma

Abstract

Assume we are given (finitely many) mutually independent variables and (finitely many) "undesirable" events, each depending on a subset of the variables of at most $k$ elements, called the scope of the event. Assume that the probability of a variable belonging to the scope of an occurring event is bounded by $q$. We prove that if $ekq \leq 1$ then there exists at least one assignment to the variables for which none of the events occurs. Examples are given where the criterion $ekq \leq 1$ is applicable, whereas that of the classical version of the Lov\'asz local lemma is not. The proof of the result is through an interactive, private-coin implementation of the algorithm by Moser. The original implementation, which yields the classical result, finds efficiently, but probabilistically, an assignment to the events that avoids all undesirable events. Interestingly, the interactive implementation given in this work does not constitute an efficient, even if probabilistic, algorithm to find an assignment as desired under the weaker assumption $ekq \leq 1$. We can only conclude that under the hypothesis that $ekq \leq 1$, the interactive protocol will produce an assignment as desired within $n$ rounds, with probability high with respect to $n$; however, the provers' choices remain non-deterministic. Plausibly finding such an assignment is inherently hard, as the situation is reminiscent, in a probabilistic framework, of problems complete for syntactic subclasses of TFNP.