Résumé
The algorithm for Lov\'{a}sz Local Lemma by Moser and Tardos gives a constructive way to prove the existence of combinatorial objects that satisfy a system of constraints. We present an alternative probabilistic analysis of the algorithm that does not involve reconstructing the history of the algorithm. We apply our technique to improve the best known upper bound to acyclic chromatic index. Specifically we show that a graph with maximum degree $\Delta$ has an acyclic proper edge coloring with at most $\lceil 3.74(\Delta-1)\rceil+1 $ colors, whereas the previously known best bound was $4(\Delta-1)$. The same technique is also applied to improve corresponding bounds for graphs with bounded girth. An interesting aspect of the latter application is that the probability of the "undesirable" events do not have a uniform upper bound, i.e., it constitutes a case of the asymmetric Lov\'{a}sz Local Lemma.