A constructive algorithm for the Lovász local lemma on permutations

While there has been significant progress on algorithmic aspects of the Lovász Local Lemma (LLL) in recent years, a noteworthy exception is when the LLL is used in the context of random permutations: the \"lopsided\" version of the LLL is usually at play here, and we do not yet have subexponential-time algorithms. We resolve this by developing a randomized polynomial-time algorithm for such applications. A noteworthy application is for Latin Transversals: the best-known general result here (Bissacot et al., improving on Erdős and Spencer), states that any n x n matrix in which each entry appears at most (27/256)n times, has a Latin transversal. We present the first polynomial-time algorithm to construct such a transversal. Our approach also yields RNC algorithms: for Latin transversals, as well as the first efficient ones for the strong chromatic number and (special cases of) acyclic edge-coloring.