An Algorithmic Proof of the Lovász Local Lemma via Resampling Oracles

Harvey, Nicholas J. A.; Vondrák, Jan

doi:10.1137/18m1167176

Cited by 12 publications

(14 citation statements)

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“…Given that all bad events can be avoided, it is often desirable to constructively find a point in the probability space (i.e., an assignment to variables in V) that avoids them. This problem has been thoroughly investigated in the sequential context [33,22,21,26,27,24,1], but somewhat less so from the point of view of parallel and distributed computation [7,13,4,6,19].…”

Section: A Gap In the Randlocal Complexity Hierarchymentioning

confidence: 99%

A Time Hierarchy Theorem for the LOCAL Model

Chang¹,

Pettie²

2019

SIAM J. Comput.

112

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The celebrated Time Hierarchy Theorem for Turing machines states, informally, that more problems can be solved given more time. The extent to which a time hierarchy-type theorem holds in the classic distributed LOCAL model has been open for many years. In particular, it is consistent with previous results that all natural problems in the LOCAL model can be classified according to a small constant number of complexities, such as Op1q, Oplog˚nq, Oplog nq, 2 Op ? log nq , etc. In this paper we establish the first time hierarchy theorem for the LOCAL model and prove that several gaps exist in the LOCAL time hierarchy. Our main results are as follows:• We define an infinite set of simple coloring problems called Hierarchical 2 1 2 -Coloring. A correctly colored graph can be confirmed by simply checking the neighborhood of each vertex, so this problem fits into the class of locally checkable labeling (LCL) problems. However, the complexity of the klevel Hierarchical 2 1 2 -Coloring problem is Θpn 1{k q, for k P Z`. The upper and lower bounds hold for both general graphs and trees, and for both randomized and deterministic algorithms.• Consider any LCL problem on bounded degree trees. We prove an automatic-speedup theorem that states that any randomized n op1q -time algorithm solving the LCL can be transformed into a deterministic Oplog nq-time algorithm. Together with a previous result [6], this establishes that on trees, there are no natural deterministic complexities in the ranges ωplog˚nq-oplog nq or ωplog nqn op1q .• We expose a gap in the randomized time hierarchy on general graphs. Roughly speaking, any randomized algorithm that solves an LCL problem in sublogarithmic time can be sped up to run in OpTLLLq time, which is the complexity of the distributed Lovász local lemma problem, currently known to be Ωplog log nq and Oplog nq.Finally, we revisit Naor and Stockmeyer's characterization of Op1q-time LOCAL algorithms for LCL problems (as order-invariant w.r.t. vertex IDs) and calculate the complexity gaps that are directly implied by their proof. For n-rings we see a ωp1q-oplog˚nq complexity gap, for p ? nˆ?nq-tori an ωp1qop a log˚nq gap, and for bounded degree trees and general graphs, an ωp1q-oplogplog˚nqq complexity gap.The goal of this paper is to understand the spectrum of natural problem complexities that can exist in the LOCAL model [31,37] of distributed computation, and to quantify the value of randomness in this model. Whereas the time hierarchy of Turing machines is known 1 to be very "smooth", recent work [6,5] has exhibited strange gaps in the LOCAL complexity hierarchy. Indeed, prior to this work it was not even known if the LOCAL model could support more than a small constant number of problem complexities. Before surveying prior work in this area, let us formally define the deterministic and randomized variants of the LOCAL model, and the class of locally checkable labeling (LCL) problems, which are intuitively those graph problems that can be computed locally in nondeterministic constant time.In bo...

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Section: A Gap In the Randlocal Complexity Hierarchymentioning

confidence: 99%

A Time Hierarchy Theorem for the LOCAL Model

Chang¹,

Pettie²

2019

SIAM J. Comput.

112

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show abstract

“…Finally, and perhaps most seriously, these algorithms are highly tailored to a single probability space. They are reminiscent of the situation for LLL algorithms before the framework of Harvey & Vondrák [23]: specialized algorithms with ad-hoc analysis.…”

Section: The Lopsided Lovász Local Lemmamentioning

confidence: 99%

“…In Harvey & Vondrák [23], the resampling oracle is presented as a randomized subroutine mapping states u ∈ B to states u ′ = R B (u). In order to define the obliviousness property, we must separate the randomness from the resampling.…”

Section: Oblivious Resampling Oraclesmentioning

confidence: 99%

“…As a starting point, we will require that r gives rise to a commutative resampling oracle with respect to a given dependence relation ∼. Before we define the new property required of the resampling function, let us reiterate the conditions of Harvey & Vondrák [23] and Kolmogorov [27], in terms of our notation.…”

Section: Oblivious Resampling Oraclesmentioning

confidence: 99%

“…Achlioptas & Iliopoulos [1] further extended this framework to include similar resampling procedures that do not directly correspond to the LLLL itself. These results have led to constructive counterparts to combinatorial results involving spanning trees, matchings of K n (both discussed in [23]) and hamiltonian cycles of K n (subsequently developed in [22]).…”

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confidence: 94%

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

Harris

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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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, B ′ 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 K n .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 K (s) n , and the first commutative resampling oracle for hamiltonian cycles of K 0 perfect matchings of K n [30], perfect matchings of the complete s-uniform hypergraph K (s) n [28], and spanning trees of K n [28].The variable-assignment setting provides one of the simplest forms of the LLLL, and the original algorithm of Moser & Tardos applies to it. In [21], Harris & Srinivasan developed an algorithm similar to the MT algorithm for the probability space of random permutations, which includes the Latin transversal application of [10]. This algorithm was very problem-specific; a more recent line of research has been developing generic LLLL algorithms, which can cover most of the probabilistic forms of the LLLL.Harvey & Vondrák [23] developed a general framework based on a "resampling oracle" R for the probability space. This is a randomized algorithm which, given some state u with some bad-event B true on u, attempts to "rerandomize" the configuration in a minimal way to fix B. This is similar to the way that the MT algorithm rerandomizes the variables involved in a bad-event. Given this resampling oracle, the following Algorithm 2 (which is a generalization of the MT algorithm) can be used...

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New bounds for the Moser‐Tardos distribution

Harris

2020

Random Struct Algorithms

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The Lovász Local Lemma (LLL) is a probabilistic principle which has been used in a variety of combinatorial constructions to show the existence of structures that have good "local" properties. In many cases, one wants more information about these structures, other than that they exist. In such case, using the "LLL-distribution", one can show that the resulting combinatorial structures have good global properties in expectation.While the LLL in its classical form is a general existential statement about probability spaces, nearly all applications in combinatorics have been turned into efficient algorithms. The simplest, variable-based setting of the LLL was covered by the seminal algorithm of Moser & Tardos (2010). This was extended by Harris & Srinivasan (2014) to random permutations, and more recently by Achlioptas & Ilioupoulos (2014) and Harvey & Vondrák (2015) to general probability spaces.One can similarly define for these algorithms an "MT-distribution," which is the distribution at the termination of the Moser-Tardos algorithm. Haeupler et al. (2011) showed bounds on the MT-distribution which essentially match the LLL-distribution for the variable-assignment setting; Harris & Srinivasan showed similar results for the permutation setting.In this work, we show new bounds on the MT-distribution which are significantly stronger than those known to hold for the LLL-distribution. In the variable-assignment setting, we show a tighter bound on the probability of a disjunctive event or singleton event. As a consequence, in k-SAT instances with bounded variable occurrence, the MT-distribution satisfies an -approximate independence condition asymptotically stronger than the LLL-distribution. We use this to show a nearly tight bound on the minimum implicate size of a CNF boolean formula. Another noteworthy application is constructing independent transversals which avoid a given subset of vertices; this provides a constructive analogue to a result of Rabern (2014).In the permutation LLL setting, we show a new type of bound which is similar to the clusterexpansion LLL criterion of Bissacot et al. (2011), but is stronger and takes advantage of the extra structure in permutations. We illustrate by showing new, stronger bounds on low-weight Latin transversals and partial Latin transversals.The Lopsided Lovász Local Lemma. Although the variable-assignment LLL is by far the most common setting in combinatorics, there are other probability spaces for which a generalized form of the LLL, known as the Lopsided Lovász Local Lemma or LLLL, applies. This was introduced by Erdős & Spencer [6], which showed that a form of the LLLL applies to the probability space defined by the uniform distribution on permutations of n letters. Erdős & Spencer used this to construct Latin transversals; the space of random permutations has since been used in a variety of other combinatorial constructions. While the space of random permutations (which we refer to as the permutation LLL) is the most well-known application of the LLLL, it also covers p...

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An Algorithmic Proof of the Lovász Local Lemma via Resampling Oracles

Cited by 12 publications

References 43 publications

A Time Hierarchy Theorem for the LOCAL Model

A Time Hierarchy Theorem for the LOCAL Model

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

New bounds for the Moser‐Tardos distribution

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