Variable-Version Lovász Local Lemma: Beyond Shearer's Bound

He, Kun; Liang, Li; Liu, Xingwu; Wang, Yuyi; Xia, Mingji

doi:10.1109/focs.2017.48

Cited by 10 publications

(49 citation statements)

References 40 publications

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“…To bound the r.h.s. of (45) we recall that ǫ ′ < 1 10 implying 3 2(2−2ǫ ′ ) < 3 2(2−1/5) = 5/6. Assuming that f (and, thus, ∆ t ) is large enough, we obtain…”

Section: E2 Proof Of Lemma E4mentioning

confidence: 99%

Focused Stochastic Local Search and the Lovász Local Lemma

Achlioptas¹,

Iliopoulos²

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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We present a new perspective on the analysis of stochastic local search algorithms via linear algebra.Our key insight is that LLL-inspired convergence arguments can be seen as a method for bounding the spectral radius of a matrix specifying the algorithm to be analyzed. Armed with this viewpoint we give a unified analysis of all entropy compression applications, connecting backtracking algorithms to the LLL in the same fashion that existing analyses connect resampling algorithms to the LLL. We then give a new convergence condition that seamlessly handles resampling algorithms that can detect, and back away from, unfavorable parts of the state space. We give several applications of this condition, notably a new vertex coloring algorithm for arbitrary graphs that uses a number of colors that matches the algorithmic barrier for random graphs. Finally, we introduce a generalization of Kolmogorov's notion of commutative algorithms [52], cast as matrix commutativity, which affords much simpler proofs both of the original results and of recent extensions.

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“…To bound the r.h.s. of (45) we recall that ǫ ′ < 1 10 implying 3 2(2−2ǫ ′ ) < 3 2(2−1/5) = 5/6. Assuming that f (and, thus, ∆ t ) is large enough, we obtain…”

Section: E2 Proof Of Lemma E4mentioning

confidence: 99%

Focused Stochastic Local Search and the Lovász Local Lemma

Achlioptas¹,

Iliopoulos²

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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“…To obtain this result, we first prove that the tight regions of CLLL and VLLL are the same for a large family of interaction bipartite graphs (see Theorem 4.8) by Bravyi and Vyalyi's Structure Lemma [7]. Then we generalize the tools for VLLL developed in [24] to CLLL, including a sufficient and necessary condition for deciding whether Shearer's bound is tight and the reduction rules. At last, we combine these tools with the conclusions for VLLL from [24] to finish the proof.…”

Section: Tight Region For Clllmentioning

confidence: 99%

“…A tight criterion under which the abstract version LLL (ALLL) holds was given by Shearer [44]. It turns out that Shearer's bound is generally not tight for variable version LLL (VLLL) [24]. Recently, Ambainis et al [3] introduced a quantum version LLL (QLLL), which was then shown to be powerful for the quantum satisfiability problem.…”

mentioning

confidence: 99%

“…The first example of gap existence is a bipartite graph whose base graph is a cycle of length 4 [28]. Recently, He et al [24] have shown that Shearer's bound is generally not tight for variable-LLL. More precisely, Shearer's bound is tight if the base graph G D is a tree, while not tight if G D has an induced cycle of length at least 4.…”

mentioning

confidence: 99%

“…A lower bound on the gap between VLLL and QLLL, especially in the symmetric direction, constitutes a quantitative analysis of the relative power of quantum. Though we have the complete characterizations of LLLs, new ideas are still needed to quantify the critical thresholds and their gaps, because the mathematical characterizations, such as Shearer's inequality system and the programm for VLLL, are usually hard to solve [22,24].…”

mentioning

confidence: 99%

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Quantum Lovász local lemma: Shearer’s bound is tight

Sun

et al. 2019

Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

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Lovász Local Lemma (LLL) is a very powerful tool in combinatorics and probability theory to show the possibility of avoiding all "bad" events under some "weakly dependent" condition. Over the last decades, the algorithmic aspect of LLL has also attracted lots of attention in theoretical computer science [23,28,35]. A tight criterion under which the abstract version LLL (ALLL) holds was given by Shearer [44]. It turns out that Shearer's bound is generally not tight for variable version LLL (VLLL) [24]. Recently, Ambainis et al. [3] introduced a quantum version LLL (QLLL), which was then shown to be powerful for the quantum satisfiability problem.In this paper, we prove that Shearer's bound is tight for QLLL, i.e., the relative dimension of the smallest satisfying subspace is completely characterized by the independent set polynomial, affirming a conjecture proposed by Sattath et al. [34,39]. Our result also shows the tightness of Gilyén and Sattath's algorithm [18], and implies that the lattice gas partition function fully characterizes quantum satisfiability for almost all Hamiltonians with large enough qudits [39].Commuting LLL (CLLL), LLL for commuting local Hamiltonians which are widely studied in the literature, is also investigated here. We prove that the tight regions of CLLL and QLLL are different in general. This result might imply that it is possible to design an algorithm for CLLL which is still efficient beyond Shearer's bound.In applications of LLLs, the symmetric cases are most common, i.e., the events are with the same probability [15,16] and the Hamiltonians are with the same relative dimension [3,39]. We give the first lower bound on the gap between the symmetric VLLL and Shearer's bound. Our result can be viewed as a quantitative study on the separation between quantum and classical constraint satisfaction problems. Additionally, we obtain similar results for the symmetric CLLL. As an application, we give lower bounds on the critical thresholds of VLLL and CLLL for several of the most common lattices.Classical Lovász Local Lemma Lovász Local Lemma (or LLL) is a very powerful tool in combinatorics and probability theory to show the possibility of avoiding all "bad" events under some "weakly dependent" condition, and has numerous applications. Formally, given a set A of bad events in a probability space, LLL provides the condition under which P(∩ A∈A A) > 0. The dependency among events is usually characterized by the dependency graph. A dependency graph is an undirected graph G D = ([m], E D ) such that for any vertex i, A i is independent of {A j : j / ∈ Γ i ∪ {i}}, where Γ i stands for the set of neighbors of i in G D . In this setting, finding the conditions under which P(∩ A∈A A) > 0 is reduced to the following problem: given a graph G D , determine its abstract interior I(G D ) which is the set of vectors p such that P ∩ A∈A A > 0 for any event set A with dependency graph G D and probability vector p. Local solutions to this problem, including the first LLL proved in 1975 by Erdős and Lovász ...

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