Parallel algorithms for the Lopsided Lovász Local Lemma

The Lovász Local Lemma (LLL) is a probabilistic tool which shows that, if a collection of “bad” events B in a probability space are not too likely and not too interdependent, then there is a positive probability that no bad-events in B occur. Moser & Tardos (2010) gave sequential and parallel algorithms which transformed most applications of the variable-assignment LLL into efficient algorithms. There has been limited success in developing parallel algorithms for more generalized forms of the LLL. Harris (2016) developed RNC algorithms for the variable-assignment Lopsided Lovász Local Lemma (LLLL) and Harris & Srinivasan (2014) developed an algorithm for the permutation LLL. These algorithms are cumbersome and limited in the types of bad-events they can handle (limitations not shared for the standard LLL). Kolmogorov (2016) developed a framework which partially parallelizes LLL settings such as random matchings of Kn, although key algorithm components remained missing. We give new parallel algorithms for most forms of LLLL, which are simpler, faster, and more general than the algorithms of Harris and Harris & Srinivasan. This also includes probability spaces for which no previous RNC algorithm was known, including matchings and hamiltonian cycles of Kn. We achieve this by providing a unified algebraic framework for applications of the LLLL to permutations. The parallel LLLL algorithm is based on new primitive for parallel computing, which we refer to as the lexicographically-first maximal-independent-set of a directed graph. We give an efficient algorithm for constructing this and show that it is precisely what is needed for our LLLL algorithms. This generalizes an algorithm given by Blelloch, Fineman, Shun (2012) for undirected graphs. We believe that this algorithm may be useful for many other types of parallel and distributed tasks. ∗Department of Computer Science, University of Maryland, College Park, MD 20742. Email: [email protected]