85 research outputs found
Improved Distributed Algorithms for the Lovász Local Lemma and Edge Coloring
The Lovász Local Lemma is a classic result in probability theory that is often used to prove the existence of combinatorial objects via the probabilistic method. In its simplest form, it states that if we have n ‘bad events’, each of which occurs with probability at most p and is independent of all but d other events, then under certain criteria on p and d, all of the bad events can be avoided with positive probability. While the original proof was existential, there has been much study on the algorithmic Lovász Local Lemma: that is, designing an algorithm which finds an assignment of the underlying random variables such that all the bad events are indeed avoided. Notably, the celebrated result of Moser and Tardos [JACM ’10] also implied an efficient distributed algorithm for the problem, running in O(log2 n) rounds. For instances with low d, this was improved to O(d 2 + logO(1) log n) by Fischer and Ghaffari [DISC ’17], a result that has proven highly important in distributed complexity theory (Chang and Pettie [SICOMP ’19]). We give an improved algorithm for the Lovász Local Lemma, providing a trade-off between the strength of the criterion relating p and d, and the distributed round complexity. In particular, in the same regime as Fischer and Ghaffari’s algorithm, we improve the round complexity to O( d log d + logO(1) log n). At the other end of the trade-off, we obtain a logO(1) log n round complexity for a substantially wider regime than previously known. As our main application, we also give the first logO(1) log n-round distributed algorithm for the problem of ∆+o(∆)-edge coloring a graph of maximum degree ∆. This is an almost exponential improvement over previous results: no prior logo(1) n-round algorithm was known even for 2∆ − 2-edge coloring
Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index
Asymptotic separation index is a parameter that measures how easily a Borel
graph can be approximated by its subgraphs with finite components. In contrast
to the more classical notion of hyperfiniteness, asymptotic separation index is
well-suited for combinatorial applications in the Borel setting. The main
result of this paper is a Borel version of the Lov\'asz Local Lemma -- a
powerful general-purpose tool in probabilistic combinatorics -- under a finite
asymptotic separation index assumption. As a consequence, we show that locally
checkable labeling problems that are solvable by efficient randomized
distributed algorithms admit Borel solutions on bounded degree Borel graphs
with finite asymptotic separation index. From this we derive a number of
corollaries, for example a Borel version of Brooks's theorem for graphs with
finite asymptotic separation index
Local Problems on Grids from the Perspective of Distributed Algorithms, Finitary Factors, and Descriptive Combinatorics
We present an intimate connection among the following fields:
(a) distributed local algorithms: coming from the area of computer science,
(b) finitary factors of iid processes: coming from the area of analysis of
randomized processes,
(c) descriptive combinatorics: coming from the area of combinatorics and
measure theory.
In particular, we study locally checkable labellings in grid graphs from all
three perspectives. Most of our results are for the perspective (b) where we
prove time hierarchy theorems akin to those known in the field (a) [Chang,
Pettie FOCS 2017]. This approach that borrows techniques from the fields (a)
and (c) implies a number of results about possible complexities of finitary
factor solutions. Among others, it answers three open questions of [Holroyd et
al. Annals of Prob. 2017] or the more general question of [Brandt et al. PODC
2017] who asked for a formal connection between the fields (a) and (b). In
general, we hope that our treatment will help to view all three perspectives as
a part of a common theory of locality, in which we follow the insightful paper
of [Bernshteyn 2020+]
Computational Hardness of Certifying Bounds on Constrained PCA Problems
Given a random n×n symmetric matrix W drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form x⊤Wx over all vectors x in a constraint set S⊂Rn. For a certain class of normalized constraint sets S we show that, conditional on certain complexity-theoretic assumptions, there is no polynomial-time algorithm certifying a better upper bound than the largest eigenvalue of W. A notable special case included in our results is the hypercube S={±1/n−−√}n, which corresponds to the problem of certifying bounds on the Hamiltonian of the Sherrington-Kirkpatrick spin glass model from statistical physics.
Our proof proceeds in two steps. First, we give a reduction from the detection problem in the negatively-spiked Wishart model to the above certification problem. We then give evidence that this Wishart detection problem is computationally hard below the classical spectral threshold, by showing that no low-degree polynomial can (in expectation) distinguish the spiked and unspiked models. This method for identifying computational thresholds was proposed in a sequence of recent works on the sum-of-squares hierarchy, and is believed to be correct for a large class of problems. Our proof can be seen as constructing a distribution over symmetric matrices that appears computationally indistinguishable from the GOE, yet is supported on matrices whose maximum quadratic form over x∈S is much larger than that of a GOE matrix.ISSN:1868-896
Global hypercontractivity and its applications
The hypercontractive inequality on the discrete cube plays a crucial role in
many fundamental results in the Analysis of Boolean functions, such as the KKL
theorem, Friedgut's junta theorem and the invariance principle. In these
results the cube is equipped with the uniform measure, but it is desirable,
particularly for applications to the theory of sharp thresholds, to also obtain
such results for general -biased measures. However, simple examples show
that when , there is no hypercontractive inequality that is strong
enough.
In this paper, we establish an effective hypercontractive inequality for
general that applies to `global functions', i.e. functions that are not
significantly affected by a restriction of a small set of coordinates. This
class of functions appears naturally, e.g. in Bourgain's sharp threshold
theorem, which states that such functions exhibit a sharp threshold. We
demonstrate the power of our tool by strengthening Bourgain's theorem, thereby
making progress on a conjecture of Kahn and Kalai and by establishing a
-biased analog of the invariance principle.
Our results have significant applications in Extremal Combinatorics. Here we
obtain new results on the Tur\'an number of any bounded degree uniform
hypergraph obtained as the expansion of a hypergraph of bounded uniformity.
These are asymptotically sharp over an essentially optimal regime for both the
uniformity and the number of edges and solve a number of open problems in the
area. In particular, we give general conditions under which the crosscut
parameter asymptotically determines the Tur\'an number, answering a question of
Mubayi and Verstra\"ete. We also apply the Junta Method to refine our
asymptotic results and obtain several exact results, including proofs of the
Huang--Loh--Sudakov conjecture on cross matchings and the
F\"uredi--Jiang--Seiver conjecture on path expansions.Comment: Subsumes arXiv:1906.0556
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