The asymptotic k-SAT threshold

Coja-Oghlan, Amin

doi:10.1145/2591796.2591822

Cited by 28 publications

(15 citation statements)

References 40 publications

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“…In this paper, however, we are concerned with the average case analysis of the Max‐Cut problem. Last decade we have seen a dramatic progress improving our understanding of various randomly generated constraint satisfaction models such as the random K‐SAT problem, the random XOR‐SAT problem, proper coloring of a random graph, independence ratio of a random graph, and many related problems . These problems broadly fall into the class of so‐called anti‐ferromagnetic spin glass models, borrowing a terminology from statistical physics.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the max‐cut of sparse random graphs

Gamarnik

2017

Random Struct Algorithms

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“…The well-known satisfiability conjecture states that for each k there exists a constant c k such that the threshold for random k-SAT is c k n. Recently a proof of this conjecture for sufficiently large values of k has been announced [13]. It is also known [11] that as k increases the threshold location scales as 2 k ln 2 − 1/2(1 + ln 2) + o k (1), thus matching to leading order the bound given by the coupon collector.…”

Section: T(x)mentioning

confidence: 99%

Speed and concentration of the covering time for structured coupon collectors

2020

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Let V be an n-set, and let X be a random variable taking values in the power-set of V. Suppose we are given a sequence of random coupons $X_1, X_2, \ldots $ , where the $X_i$ are independent random variables with distribution given by X. The covering time T is the smallest integer $t\geq 0$ such that $\bigcup_{i=1}^t X_i=V$ . The distribution of T is important in many applications in combinatorial probability, and has been extensively studied. However the literature has focused almost exclusively on the case where X is assumed to be symmetric and/or uniform in some way.In this paper we study the covering time for much more general random variables X; we give general criteria for T being sharply concentrated around its mean, precise tools to estimate that mean, as well as examples where T fails to be concentrated and when structural properties in the distribution of X allow for a very different behaviour of T relative to the symmetric/uniform case.

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“…For k = 2, the location of the transition is known exactly to be r 2 = 1 [10]. For k ≥ 3, bounds asymptotic in k [11] and exact results for large constant k [16] are now known. The phenomenon of a sharp solubility transition is also interesting from the perspective of computational complexity and algorithm engineering, since it appears to coincide with a regime of formulas that are particularly difficult to solve by complete SAT solvers [31].…”

Section: The Solubility Phase Transitionmentioning

confidence: 99%

On the Empirical Time Complexity of Scale-Free 3-SAT at the Phase Transition

Bläsius

Friedrich

Sutton

2019

Tools and Algorithms for the Construction and Analysis of Systems

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The hardness of formulas at the solubility phase transition of random propositional satisfiability (SAT) has been intensely studied for decades both empirically and theoretically. Solvers based on stochastic local search (SLS) appear to scale very well at the critical threshold, while complete backtracking solvers exhibit exponential scaling. On industrial SAT instances, this phenomenon is inverted: backtracking solvers can tackle large industrial problems, where SLS-based solvers appear to stall. Industrial instances exhibit sharply different structure than uniform random instances. Among many other properties, they are often heterogeneous in the sense that some variables appear in many while others appear in only few clauses. We conjecture that the heterogeneity of SAT formulas alone already contributes to the trade-off in performance between SLS solvers and complete backtracking solvers. We empirically determine how the run time of SLS vs. backtracking solvers depends on the heterogeneity of the input, which is controlled by drawing variables according to a scale-free distribution. Our experiments reveal that the efficiency of complete solvers at the phase transition is strongly related to the heterogeneity of the degree distribution. We report results that suggest the depth of satisfying assignments in complete search trees is influenced by the level of heterogeneity as measured by a power-law exponent. We also find that incomplete SLS solvers, which scale well on uniform instances, are not affected by heterogeneity. The main contribution of this paper utilizes the scale-free random 3-SAT model to isolate heterogeneity as an important factor in the scaling discrepancy between complete and SLS solvers at the uniform phase transition found in previous works.

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The asymptotic k-SAT threshold

Cited by 28 publications

References 40 publications

On the max‐cut of sparse random graphs

On the max‐cut of sparse random graphs

Speed and concentration of the covering time for structured coupon collectors

On the Empirical Time Complexity of Scale-Free 3-SAT at the Phase Transition

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