A Constructive Lovász Local Lemma for 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 breakthrough algorithm of Moser & Tardos only works in the setting of independent variables, and does not apply in this context. We resolve this by developing a randomized polynomial-time algorithm for such applications. A noteworthy application is for Latin transversals: the best general result known here (Bissacot et al., improving on Erdős and Spencer), states that any n×n matrix in which each entry appears at most (27/256)n times, has a Latin transversal. We present the first polynomialtime algorithm to construct such a transversal. We also develop RNC algorithms for Latin transversals, rainbow Hamiltonian cycles, strong chromatic number, and hypergraph packing. In addition to efficiently finding a configuration which avoids bad events, the algorithm of Moser & Tardos has many powerful extensions and properties. These include a well-characterized distribution on the output distribution, parallel algorithms, and a partial ∗An extended abstract of this paper has appeared in the Proceedings of the 25th ACM-SIAM Symposium on Discrete Algorithms, 2014 [19]. †Research supported in part by NSF Awards CNS-1010789 and CCF-1422569. ‡Research supported in part by NSF Awards CNS-1010789 and CCF-1422569, and by a research award from Adobe, Inc. ACM Classification: F.2.2, G.3 AMS Classification: 68W20, 60C05, 05B15