2017/08/13 by Lefteris M. Kirousis, Lefteris Kirousis, John Livieratos +4
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Advanced Graph Theory Research #Computer science #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Lemma (botany) #Mathematics #cs.DM #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1708.03954
Mistakes were found
openalex publication_date 2017/08/13 · arxiv created 2018/08/07 · arxiv updated 2018/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ≤ 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 ≤ 1 is applicable, whereas that of the classical version of the Lovász 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 ≤ 1. We can only conclude that under the hypothesis that ekq ≤ 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.