Towards the sampling Lovász Local Lemma

Jain, Vishesh; Pham, Huy Tuan; Vuong, Thuy Duong

doi:10.1109/focs52979.2021.00025

Cited by 10 publications

(11 citation statements)

References 14 publications

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“…Instead, we apply a more average-case style analysis. Interestingly, some of our analysis and the LLL condition resemble those in [JPV21b] for a deterministic approximate counting algorithm with time complexity poly ( ,Δ,log ) .…”

Section: Marginal Samplermentioning

confidence: 81%

“…e only loose end now is that the LLL condition is not self-reducible, meaning it is not invariant under arbitrary pinning. We adopt the idea of "freezing" constraints used in [JPV21b] to guide the algorithm to pick variables for sampling.…”

Section: Marginal Samplermentioning

confidence: 99%

“…e main algorithm (Algorithm 1) is the same as the main sampling frameworks in [JPV21b,GLLZ19]. A partial assignment ∈ Q is maintained, initially as the empty assignment = ⋆ ⋆ .…”

Section: 1mentioning

confidence: 99%

“…In a seminal work of Moitra [Moi19], a very innovative algorithm was given for sampling almost uniform -CNF solutions assuming an LLL condition Δ 60 1. is sampling algorithm was based on deterministic approximate counting by solving linear programs on properly factorized formulas and has a running time of poly ( ,Δ) . is LP-based approach was later extended to hypergraph coloring [GLLZ19] and random CNF formulas [GGGY20], and finally in a work of Jain, Pham and Vuong [JPV21b] to all CSPs satisfying a substantially improved LLL condition Δ 7 1. All these deterministic approximate counting based algorithms suffered from an poly ( ,Δ) time cost.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Sampling Lovász Local Lemma For General Constraint Satisfaction Solutions In Near-Linear Time

He¹,

Wang²,

Yin³

2022

Preprint

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A. We give a fast algorithm for sampling uniform solutions of general constraint satisfaction problems (CSPs) in a local lemma regime. e expected running time of our algorithm is near-linear in and a fixed polynomial in Δ, where is the number of variables and Δ is the max degree of constraints. Previously, up to similar conditions, sampling algorithms with running time polynomial in both and Δ, only existed for the almost atomic case, where each constraint is violated by a small number of forbidden local configurations.Our sampling approach departs from all previous fast algorithms for sampling LLL, which were based on Markov chains. A crucial step of our algorithm is a recursive marginal sampler that is of independent interests. Within a local lemma regime, this marginal sampler can draw a random value for a variable according to its marginal distribution, at a local cost independent of the size of the CSP.the algorithm terminates within poly( , , Δ)• log time in expectation and outputs an almost uniform sample of satisfying assignments for Φ within total variation distance.e formal statement of the theorem is in eorem 5.1 (for termination and correctness of sampling) and in eorem 6.3 (for efficiency of sampling).e condition in (3) becomes Δ 7+ (1) 1 when ≤ ( ) − (1) , while a typical case is usually given by a much smaller ≤ −Ω ( ) .e previous best bound for sampling general CSP solutions was that 3 Δ 7 < for a small constant , achieved by the deterministic approximate counting based algorithm in [JPV21b] whose running time was ( / ) poly ( ,Δ,log ) .Let be the total number of satisfying assignments for Φ. A ˆ is called an -approximation of if (1 − ) ≤ ˆ ≤ (1 + ) . By routinely going through the non-adaptive annealing process in [FGYZ21], the approximate sampler in eorem 1.1 can be used as a black-box to give for any ∈ (0, 1) an -approximation of in time poly ( , , Δ) • ˜ 2 −2 with high probability.1.1.1. Perfect sampler.e evaluation oracle in Assumption 1 in fact checks the sign of P[¬ | ], the probability that a constraint is violated given a partially specified assignment . If further this probability can be estimated efficiently, then the sampling in eorem 1.1 can be made perfect, where the output sample follows exactly the target distribution.

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Section: Marginal Samplermentioning

confidence: 81%

Section: Marginal Samplermentioning

confidence: 99%

“…e main algorithm (Algorithm 1) is the same as the main sampling frameworks in [JPV21b,GLLZ19]. A partial assignment ∈ Q is maintained, initially as the empty assignment = ⋆ ⋆ .…”

Section: 1mentioning

confidence: 99%

mentioning

confidence: 99%

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Sampling Lovász Local Lemma For General Constraint Satisfaction Solutions In Near-Linear Time

He¹,

Wang²,

Yin³

2022

Preprint

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“…Moitra's algorithm relies on the celebrated Lovász local lemma and is later referred to as counting/sampling Lovász local lemma. Since then, the study of counting and sampling solutions of bounded-degree formulas has been fruitful, including: hardness result [BGG + 19, GGW21], k-CNF formulas [GJL19, Moi19, Har20, FGYZ21, FHY21, QWZ22], hypergraph coloring [GLLZ19, FHY21, FGW22], general atomic setting [JPV21a,HSW21], and general non-atomic setting [JPV21b,HWY22b,HWY22a]. The current best bound of counting/sampling solutions of bounded-degree k-CNF formulas is d 2 k/5 [HWY22a].…”

Section: Introductionmentioning

confidence: 99%

Improved Bounds for Sampling Solutions of Random CNF Formulas

He¹,

Wu²,

Yang³

2022

Preprint

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Let Φ be a random k-CNF formula on n variables and m clauses, where each clause is a disjunction of k literals chosen independently and uniformly. Our goal is, for most Φ, to (approximately) uniformly sample from its solution space.Let α = m/n be the density. The previous best algorithm runs in time n poly(k,α) for any α 2 k/300 [Galanis, Goldberg, Guo, and Yang, SIAM J. Comput.'21]. In contrast, our algorithm runs in near-linear time for any α 2 k/3 .

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On the zeroes of hypergraph independence polynomials

Galvin,

McKinley,

Perkins

et al. 2023

Combinator. Probab. Comp.

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We study the locations of complex zeroes of independence polynomials of bounded-degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer’s result on the optimal zero-free disc, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree $\Delta$ have a zero-free disc almost as large as the optimal disc for graphs of maximum degree $\Delta$ established by Shearer (of radius $\sim 1/(e \Delta )$ ). Up to logarithmic factors in $\Delta$ this is optimal, even for hypergraphs with all edge sizes strictly greater than $2$ . We conjecture that for $k\ge 3$ , $k$ -uniform linear hypergraphs have a much larger zero-free disc of radius $\Omega (\Delta ^{- \frac{1}{k-1}} )$ . We establish this in the case of linear hypertrees.

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Towards the sampling Lovász Local Lemma

Cited by 10 publications

References 14 publications

Sampling Lovász Local Lemma For General Constraint Satisfaction Solutions In Near-Linear Time

Sampling Lovász Local Lemma For General Constraint Satisfaction Solutions In Near-Linear Time

Improved Bounds for Sampling Solutions of Random CNF Formulas

On the zeroes of hypergraph independence polynomials

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