40 research outputs found
Commutative Algorithms Approximate the LLL-distribution
Following the groundbreaking Moser-Tardos algorithm for the Lovasz Local
Lemma (LLL), a series of works have exploited a key ingredient of the original
analysis, the witness tree lemma, in order to: derive deterministic, parallel
and distributed algorithms for the LLL, to estimate the entropy of the output
distribution, to partially avoid bad events, to deal with super-polynomially
many bad events, and even to devise new algorithmic frameworks. Meanwhile, a
parallel line of work, has established tools for analyzing stochastic local
search algorithms motivated by the LLL that do not fall within the Moser-Tardos
framework. Unfortunately, the aforementioned results do not transfer to these
more general settings. Mainly, this is because the witness tree lemma,
provably, no longer holds. Here we prove that for commutative algorithms, a
class recently introduced by Kolmogorov and which captures the vast majority of
LLL applications, the witness tree lemma does hold. Armed with this fact, we
extend the main result of Haeupler, Saha, and Srinivasan to commutative
algorithms, establishing that the output of such algorithms well-approximates
the LLL-distribution, i.e., the distribution obtained by conditioning on all
bad events being avoided, and give several new applications. For example, we
show that the recent algorithm of Molloy for list coloring number of sparse,
triangle-free graphs can output exponential many list colorings of the input
graph
Stochastic Control via Entropy Compression
We consider an agent trying to bring a system to an acceptable state by
repeated probabilistic action. Several recent works on algorithmizations of the
Lovasz Local Lemma (LLL) can be seen as establishing sufficient conditions for
the agent to succeed. Here we study whether such stochastic control is also
possible in a noisy environment, where both the process of state-observation
and the process of state-evolution are subject to adversarial perturbation
(noise). The introduction of noise causes the tools developed for LLL
algorithmization to break down since the key LLL ingredient, the sparsity of
the causality (dependence) relationship, no longer holds. To overcome this
challenge we develop a new analysis where entropy plays a central role, both to
measure the rate at which progress towards an acceptable state is made and the
rate at which noise undoes this progress. The end result is a sufficient
condition that allows a smooth tradeoff between the intensity of the noise and
the amenability of the system, recovering an asymmetric LLL condition in the
noiseless case.Comment: 18 page
Improved Bounds for Coloring Locally Sparse Hypergraphs
We show that, for every k ? 2, every k-uniform hypergaph of degree ? and girth at least 5 is efficiently (1+o(1))(k-1) (? / ln ?)^{1/(k-1)}-list colorable. As an application we obtain the currently best deterministic algorithm for list-coloring random hypergraphs of bounded average degree
LIPIcs
The Lovász Local Lemma (LLL) is a powerful tool in probabilistic combinatorics which can be used to establish the existence of objects that satisfy certain properties. The breakthrough paper of Moser and Tardos and follow-up works revealed that the LLL has intimate connections with a class of stochastic local search algorithms for finding such desirable objects. In particular, it can be seen as a sufficient condition for this type of algorithms to converge fast. Besides conditions for existence of and fast convergence to desirable objects, one may naturally ask further questions regarding properties of these algorithms. For instance, "are they parallelizable?", "how many solutions can they output?", "what is the expected "weight" of a solution?", etc. These questions and more have been answered for a class of LLL-inspired algorithms called commutative. In this paper we introduce a new, very natural and more general notion of commutativity (essentially matrix commutativity) which allows us to show a number of new refined properties of LLL-inspired local search algorithms with significantly simpler proofs