An interactive version of Lovász local lemma: Arthur and Merlin implement Moser's algorithm
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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, known as the scope of the event. Assume that the probability of an individual variable belonging to the scope of an occurring event is bounded by q. We prove that if ekq ≤ 1 then there exists at least one assignment to the variables for which none of the events occurs. This result is stronger than the classical version of the Lovász local lemma, which is expressed in terms of a bound p of the probabilities of the individual events, and of d, a bound on the degree of the dependency graph. The proof is through a public coin, interactive 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 ≤ 1. We can only conclude that under this hypothesis, the interactive protocol will produce an assignment as desired within n rounds, with probability high with respect to n; however, the provers' (Merlin's) 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.
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