Oblivious resampling oracles and parallel algorithms for the Lopsided Lovász Local Lemma

Abstract

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. A framework of Harvey & Vondrák (2015) based on “resampling oracles” extended this give very general sequential algorithms for other probability spaces satisfying the Lopsided Lovász Local Lemma (LLLL). We describe a new structural property of resampling oracles which holds for all known resampling oracles, which we call “obliviousness.” Essentially, it means that the interaction between two bad-events B, B0 depends only on the randomness used to resample B, and not on the precise state within B itself. This property has two major consequences. First, it is the key to achieving a unified parallel LLLL algorithm, which is faster than previous, problem-specific algorithms of Harris (2016) for the variable-assignment LLLL algorithm and of Harris & Srinivasan (2014) for permutations. This new algorithm extends a framework of Kolmogorov (2016), and gives the first RNC algorithms for rainbow perfect matchings and rainbow hamiltonian cycles of Kn. Second, this property allows us to build LLLL probability spaces out of a relatively simple “atomic” set of events. It was intuitively clear that existing LLLL spaces were built in this way; but the obliviousness property formalizes this and gives a way of automatically turning a resampling oracle for atomic events into a resampling oracle for conjunctions of them. Using this framework, we get the first sequential resampling oracle for rainbow perfect matchings on the complete s-uniform hypergraph Kn(s), and the first commutative resampling oracle for hamiltonian cycles of Kn