Satisfiability Threshold for Random Regular nae-sat

Ding, Jian; Sly, Allan; Sun, Nike

doi:10.1007/s00220-015-2492-8

Cited by 26 publications

(22 citation statements)

References 31 publications

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“…For problems such as k-NAESAT, k-XORSAT or graph coloring where the first moment provides the correct answer due to inherent symmetry properties, the second moment method and small subgraph conditioning yield very precise information as to the number of solutions [16,19,45]. Verifying that the number of solutions is determined by the physicists" 1RSB formula [35], the contribution of Sly, Sun and Zhang [48] on the random regular k-NAESAT problem near its satisfiability threshold [25] deals with an even more intricate scenario.…”

Section: Discussionmentioning

confidence: 99%

The number of satisfying assignments of random 2‐SAT formulas

Achlioptas

Coja-Oghlan

Hahn‐Klimroth

et al. 2021

Random Struct Algorithms

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We show that throughout the satisfiable phase the normalized number of satisfying assignments of a random 2-SAT formula converges in probability to an expression predicted by the cavity method from statistical physics. The proof is based on showing that the Belief Propagation algorithm renders the correct marginal probability that a variable is set to "true" under a uniformly random satisfying assignment. KEYWORDS 2-SAT, Belief Propagation, satisfiability problem 1 INTRODUCTION Background and motivationThe random 2-SAT problem was the first random constraint satisfaction problem whose satisfiability threshold could be pinpointed precisely, an accomplishment attained independently by Chvátal and Reed [15] and Goerdt [31] in 1992. The proofs evince the link between the 2-SAT threshold and the percolation phase transition of a random digraph. This connection subsequently enabled Bollobás, Borgs, Chayes, Kim, and Wilson [12] to identify the size of the scaling window, which matches thatThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Section: Discussionmentioning

confidence: 99%

The number of satisfying assignments of random 2‐SAT formulas

Achlioptas

Coja-Oghlan

Hahn‐Klimroth

et al. 2021

Random Struct Algorithms

View full text Add to dashboard Cite

show abstract

“…We remark that in [33], they introduced a truncation of free and red variables, but this truncation only induces a difference that is exponentially small in n (see Lemma 2.12 of [33], or Lemma 3.3 of [39]). Thus, taking expectation in (18) shows that…”

Section: Recall the Notation Of Coloring Profilementioning

confidence: 99%

“…In a typical solution, a small but constant fraction of variables can be flipped between 0 and 1 without violating any constraints giving rise to exponentially many nearby solutions. In order give a combinatorial definition of a cluster, the so-called coarsening algorithm inductively maps variables taking values in {0, 1} to f, called a free variable, if they can be flipped without violating any constraints [36,18]. A constraint is considered satisfied if one of its variables is free and the algorithm continues until no more variables can be set to f resulting in a {0, 1, f} valued configuration called a frozen configuration.…”

Section: Clustersmentioning

confidence: 99%

“…We call x ∈ {0, 1, f} V a (valid) frozen configuration if it satisfies Frozen model solutions are themselves a random csp but without cluster, because they are in effect clusters of solutions projected to a single point. [18] showed that for α ∈ [α lbd , α ubd ], with high probability all solutions map via coarsening algorithms to frozen configurations with a low density (less than 7 2 k ) of free variables. Thus, free variables form subcrtical clusters that are almost all small subcritical branching process trees.…”

Section: Clustersmentioning

confidence: 99%

“…From the mathematics literature, a great deal of progress has been made in understanding the satisfiability threshold α sat for k-sat [16,19], nae-sat [14,18] and coloring models [3,10]. Once we understand for what range of constraint densities solutions exist, it is natural to ask more detailed questions about the solutions, such as how many there are and what is the behavior of a typical solution.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Number of Solutions for Random Regular NAE-SAT

Sly

Sun

Zhang

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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Recent work has made substantial progress in understanding the transitions of random constraint satisfaction problems. In particular, for several of these models, the exact satisfiability threshold has been rigorously determined, confirming predictions of statistical physics. Here we revisit one of these models, random regular k-nae-sat: knowing the satisfiability threshold, it is natural to study, in the satisfiable regime, the number of solutions in a typical instance. We prove here that these solutions have a well-defined free energy (limiting exponential growth rate), with explicit value matching the one-step replica symmetry breaking prediction. The proof develops new techniques for analyzing a certain "survey propagation model" associated to this problem. We believe that these methods may be applicable in a wide class of related problems.

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On the max‐cut of sparse random graphs

Gamarnik

2017

Random Struct Algorithms

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We consider the problem of estimating the size of a maximum cut (Max-Cut problem) in a random Erdős-Rényi graph on n nodes and ⌊cn⌋ edges. It is shown in Coppersmith et al. [CGHS04] that the size of the maximum cut in this graph normalized by the number of nodes belongs to the asymptotic region [c/2 + 0.37613 √ c, c/2 + 0.58870 √ c] with high probability (w.h.p.) as n increases, for all sufficiently large c. The upper bound was obtained by application of the first moment method, and the lower bound was obtained by constructing algorithmically a cut which achieves the stated lower bound.In this paper we improve both upper and lower bounds by introducing a novel bounding technique. Specifically, we establish that the size of the maximum cut normalized by the number of nodes belongs to the interval [c/2 + 0.47523 √ c, c/2 + 0.55909 √ c] w.h.p. as n increases, for all sufficiently large c. Instead of considering the expected number of cuts achieving a particular value as is done in the application of the first moment method, we observe that every maximum size cut satisfies a certain local optimality property, and we compute the expected number of cuts with a given value satisfying this local optimality property. Estimating this expectation amounts to solving a rather involved two dimensional large deviations problem. We solve this underlying large deviation problem asymptotically as c increases and use it to obtain an improved upper bound on the Max-Cut value. The lower bound is obtained by application of the second moment method, coupled with the same local optimality constraint, and is shown to work up to the stated lower bound value c/2+0.47523 √ c. It is worth noting that both bounds are stronger than the ones obtained by standard first and second moment methods.Finally, we also obtain an improved lower bound of 1.36000n on the Max-Cut for the random cubic graph or any cubic graph with large girth, improving the previous best bound of 1.33773n.

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Satisfiability Threshold for Random Regular nae-sat

Cited by 26 publications

References 31 publications

The number of satisfying assignments of random 2‐SAT formulas

The number of satisfying assignments of random 2‐SAT formulas

The Number of Solutions for Random Regular NAE-SAT

On the max‐cut of sparse random graphs

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