Catching the k-NAESAT threshold

Coja-Oglan, Amin; Παναγιώτου, Κωνσταντίνος

doi:10.1145/2213977.2214058

Cited by 48 publications

(75 citation statements)

References 42 publications

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“…In [10] we obtained a result similar to Theorem 1 for the (symmetric) random k-NAESAT problem. In symmetric problems many of the maneuvers that we are going to have to go through (e.g., clause/variable types, see Section 4) are unnecessary.…”

Section: Related Worksupporting

confidence: 59%

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The asymptotic k-SAT threshold

Coja-Oghlan

2014

Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing

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Since the early 2000s physicists have developed an ingenious but non-rigorous formalism called the cavity method to put forward precise conjectures as to the phase transitions in random constraint satisfaction problems ("CSPs"). The cavity method comes in two versions: the simpler replica symmetric variant, and the more intricate 1-step replica symmetry breaking ("1RSB") version. While typically the former only gives upper and lower bounds, the latter is conjectured to yield precise results in many cases. By now, there are a number of examples where the replica symmetric bounds have been verified rigorously. However, verifications of 1RSB predictions are scarce. Perhaps the most prominent challenge in this context is that of pinning down the random k-SAT threshold rk−SAT. Here we prove that r k−SAT = 2 k ln 2 − 1 2 (1 + ln 2) + o k (1), which matches the 1RSB prediction up to the o k (1) error term. The proof directly employs ideas from the 1RSB cavity method, such as the notion of covers (relaxed satisfying assignments) and bits of the Survey Propagation calculations. The best previous lower bound was r k−SAT ≥ 2 k ln 2 − 3 2 ln 2 + o k (1), matching the replica symmetric lower bound asymptotically [Coja-Oghlan, Panagiotou: STOC 2013].

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Section: Related Worksupporting

confidence: 59%

“…Both of these problems are symmetric. The proofs in [14,15] are based on the second moment method applied to a notion of "cover" appropriate for NAESAT/independent sets, while [10] relies on an ad-hoc concept called "heavy solutions".…”

Section: Related Workmentioning

confidence: 99%

The asymptotic k-SAT threshold

Coja-Oghlan

2014

Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing

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“…Moreover, the best current bounds on the random (hyper)graph k-colorability thresholds are based on "vanilla" second moment arguments as well [5,15]. In summary, in all the previous second moment arguments, the issue of asymmetry either did not appear at all by the nature of the problem [4,5,11,12,14,15,19], or it was sidestepped [6].…”

Section: Related Workmentioning

confidence: 99%

“…Perhaps most importantly, in most random CSPs the threshold for the existence of solutions is not known precisely. In the relatively simple case of the random k-NAESAT ("NotAll-Equal-Satisfiability") problem the difference between the best current lower and upper bounds is as tiny as 2 −Ω(k) [11]. By contrast, in random graph k-coloring, a problem already studied by Erdős and Rényi in the 1960s, the best current bounds differ by Θ(ln k) [5].…”

Section: Introductionmentioning

confidence: 99%

Going after the k-SAT threshold

Coja-Oghlan

Παναγιώτου²

2013

Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing

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Random k-SAT is the single most intensely studied example of a random constraint satisfaction problem. But despite substantial progress over the past decade, the threshold for the existence of satisfying assignments is not known precisely for any k ≥ 3. The best current results, based on the second moment method, yield upper and lower bounds that differ by an additive k · ln 2/2, a term that is unbounded in k (Achlioptas, Peres: STOC 2003). The basic reason for this gap is the inherent asymmetry of the Boolean values 'true' and 'false' in contrast to the perfect symmetry, e.g., among the various colors in a graph coloring problem. Here we develop a new asymmetric second moment method that allows us to tackle this issue head on for the first time in the theory of random CSPs. This technique enables us to compute the k-SAT threshold up to an additive ln 2 − 1 2 + O(1/k) ≈ 0.19. Independently of the rigorous work, physicists have developed a sophisticated but non-rigorous technique called the "cavity method" for the study of random CSPs (Mézard, Parisi, Zecchina: Science 2002). Our result matches the best bound that can be obtained from the so-called "replica symmetric" version of the cavity method, and indeed our proof directly harnesses parts of the physics calculations.

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“…They explain many phenomena, most notably why some random CSP's are algorithmically very challenging (eg, [1,18]). Intuition gained from these hypotheses has led to some very impressive heuristics (eg, Survey Propogation [4,30,32], and the best of the random r-SAT algorithms whose performance has been rigorously proven [10]), and some remarkably tight rigorous bounds on various satisfiability thresholds [11][12][13][14]. Ding, Sly and Sun recently used an approach outlined by these hypotheses to prove the k-SAT conjecture [19], with a determination of the k-SAT satisfiability threshold for all large k. It is clear that, in order to approach many of the outstanding challenges around random CSP's, we need to understand clustering.…”

Section: Introductionmentioning

confidence: 99%

Inside the clustering window for random linear equations

Gao

Molloy

2017

Random Struct Algorithms

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We study a random system of cn linear equations over n variables in GF(2), where each equation contains exactly r variables; this is equivalent to r‐XORSAT. Previous work has established a clustering threshold, cr∗ for this model: if c=cr∗−ϵ for any constant ϵ>0 then with high probability all solutions form a well‐connected cluster; whereas if c=cr∗+ϵ, then with high probability the solutions partition into well‐connected, well‐separated clusters (with probability tending to 1 as n→∞). This is part of a general clustering phenomenon which is hypothesized to arise in most of the commonly studied models of random constraint satisfaction problems, via sophisticated but mostly nonrigorous techniques from statistical physics. We extend that study to the range c=cr∗+o(1), and prove that the connectivity parameters of the r‐XORSAT clusters undergo a smooth transition around the clustering threshold.

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Catching the k-NAESAT threshold

Cited by 48 publications

References 42 publications

The asymptotic k-SAT threshold

The asymptotic k-SAT threshold

Going after the k-SAT threshold

Inside the clustering window for random linear equations

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