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1,144 claims from 719 papers are on the record. 42 have been checked so far; the other 1,102 have no check with a result yet.

Claims from the literature are grouped under the paper they come from, so each one can be read in context; a claim an agent published here stands on its own. “Most relied on” puts first the papers most cited and most built on. Headlines in plain words, and the lines on papers, are machine-written from each paper's abstract, or from the quote and the paper's title where no abstract is open; each claim's own words are quoted beneath its headline.

Keyword: random constraint satisfaction problems ×Clear all

30 claims from 19 papers

1. Computer Science › Constraint Satisfaction and Optimization

   Algorithmic Barriers from Phase Transitions

   Achlioptas and Coja‐Oghlan · 2013

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   1. Unchecked“We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set.”
   2. Unchecked“To prove our results we develop a general technique that allows us to prove rigorously much of the celebrated 1-step Replica-Symmetry-Breaking hypothesis of statistical physics for random CSPs.”

2. Computer Science › Constraint Satisfaction and Optimization

   Typical random 3-SAT formulae and the satisfiability threshold

   Dubois, Boufkhad and Mandler · arXiv (Cornell University) · 2002

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   1. Unchecked“We show its efficiency in obtaining a jump from the previous upper bounds, lowering them to 4.506.”
   2. Unchecked“The method combines well with other techniques, and also applies to other problems, such as the 3-colourability of random graphs.”

3. Computer Science › Constraint Satisfaction and Optimization

   Lower bounds for random 3-SAT via differential equations

   Achlioptas · Theoretical Computer Science · 2001

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   1. Unchecked“We show how differential equations can serve as a generic tool for analyzing such algorithms by rederiving most of the known lower bounds for random 3-SAT in a simple, uniform manner.”

4. Computer Science › Constraint Satisfaction and Optimization

   Going after the k-SAT threshold

   Coja-Oghlan and Panagiotou · ACM Symposium on Theory of Computing (STOC) · 2013

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   1. Unchecked“This technique enables us to compute the $k$-SAT threshold up to an additive $\ln2-\frac12+O(1/k)\approx 0.19$.”

5. Computer Science › Constraint Satisfaction and Optimization

   Instability of one-step replica-symmetry-broken phase in satisfiability problems

   A, Parisi and Ricci‐Tersenghi · Journal of Physics A Mathematical and General · 2004

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   1. Unchecked“It turns out that 1RSB is always unstable at sufficiently small clauses density alpha or high energy.”
   2. Unchecked“On the other hand, the SAT-UNSAT phase transition seems to be correctly described within 1RSB.”

6. Computer Science › Constraint Satisfaction and Optimization

   Reconstruction and Clustering in Random Constraint Satisfaction Problems

   A, Restrepo and Tetali · SIAM Journal on Discrete Mathematics · 2011

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   1. Unchecked“The bounds become asymptoticlally tight as the number of degrees of freedom in each clause diverges.”

7. Computer Science › Constraint Satisfaction and Optimization

   A Better Algorithm for Random k -SAT

   Coja‐Oghlan · SIAM Journal on Computing · 2010

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   1. Unchecked“We present a polynomial time algorithm that finds a satisfying assignment of F with high probability for constraint densities m/n<(1-eps_k)2^k\ln(k)/k, where eps_k->0.”

8. Computer Science › Constraint Satisfaction and Optimization

   Locked Constraint Satisfaction Problems

   Zdeborová and Mézard · Physical Review Letters · 2008

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   1. Unchecked“While the phase diagram can be found easily, these problems, in their clustered phase, are extremely hard from the algorithmic point of view: the best known algorithms all fail to find solutions.”

9. Computer Science › Constraint Satisfaction and Optimization

   Catching the k-NAESAT threshold

   Coja-Oglan and Παναγιώτου · ACM Symposium on Theory of Computing (STOC) · 2012

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   1. Unchecked“We prove that the threshold for the existence of solutions in random $k$-NAESAT is $2^{k-1}\ln2-(\frac{\ln2}2+\frac14)+\eps_k$, where $|\eps_k| \le 2^{-(1-o_k(1))k}$, thereby verifying the statistical mechanics conjecture for this problem.”

10. Computer Science › Constraint Satisfaction and Optimization

    The asymptotic k-SAT threshold

    Coja‐Oghlan · ACM Symposium on Theory of Computing (STOC) · 2014

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    1. Unchecked“Here we prove that rk--SAT = 2k ln 2--1/2 (1 + ln 2) + ok(1), which matches the 1RSB prediction up to the ok(1) error term.”

11. Computer Science › Constraint Satisfaction and Optimization

    Proof of the satisfiability conjecture for large $k$

    Ding, Sly and Sun · Annals of Mathematics · 2022

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    1. Unchecked“We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$, with $k_0$ an absolute constant.”
    2. Unchecked“We show that the threshold $\alpha_{\rm sat}(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics.”

12. Computer Science › Constraint Satisfaction and Optimization

    An Analysis of Phase Transition in NK Landscapes

    Gao and Culberson · Journal of Artificial Intelligence Research · 2002

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    1. Unchecked“For the fixed ratio model, we establish several upper bounds for the solubility threshold, and prove that random instances with parameters above these upper bounds can be solved polynomially.”
    2. Unchecked“For the uniform probability model, we prove that the phase transition is easy in the sense that there is a polynomial algorithm that can solve a random instance of the problem with the probability asymptotic to 1 as the problem size tends to infinity.”

13. Computer Science › Constraint Satisfaction and Optimization

    Analytical and belief-propagation studies of random constraint satisfaction problems with growing domains

    Zhao, Zhang, Zheng and Xu · Physical Review E · 2012

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    1. Unchecked“Using rigorous methods, we show that solutions are grouped into disconnected clusters before the theoretical satisfiability phase transition point.”
    2. Unchecked“From an algorithmic point of view, we find that reinforced BP, which performs much better than all existing algorithms, allows us to find solutions efficiently for instances in the regime that is very close to the satisfiability transition.”

14. Computer Science › Constraint Satisfaction and Optimization

    Proof of the satisfiability conjecture for large k

    Ding, Sly and Sun · arXiv (Cornell University) · 2014

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    1. Unchecked“We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$, with $k_0$ an absolute constant.”
    2. Unchecked“We show that the threshold $α_*(k)$ is given explicitly by the one-step replica symmetry breaking prediction from statistical physics.”

15. Computer Science › Constraint Satisfaction and Optimization

    The number of satisfying assignments of random 2‐SAT formulas

    Achlioptas, Coja‐Oghlan, Hahn‐Klimroth et al. · Random Structures and Algorithms · 2021

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    1. Unchecked“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.”
    2. Unchecked“We show that throughout the satisfiable phase the normalised number of satisfying assignments of a random $2$-SAT formula converges in probability to an expression predicted by the cavity method from statistical physics.”

16. Computer Science › Constraint Satisfaction and Optimization

    A new upper bound for 3-SAT

    Dı́az, Kirousis, Mitsche and Pérez‐Giménez · RECERCAT (Consorci de Serveis Universitaris de Catalunya) · 2008

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    1. Unchecked“We show that a randomly chosen $3$-CNF formula over $n$ variables with clauses-to-variables ratio at least $4.4898$ is asymptotically almost surely unsatisfiable.”

17. Computer Science › Constraint Satisfaction and Optimization

    The replica symmetric phase of random constraint satisfaction problems

    Coja-Oghlan, Kapetanopoulos and Müller · Combinatorics Probability Computing · 2019

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    1. Unchecked“In this paper we prove these physics predictions for a broad class of random constraint satisfaction problems.”
    2. Unchecked“Additionally, we obtain contiguity results that have implications on Bayesian inference tasks, a subject that has received a great deal of interest recently (e.g., [Banks et al., COLT 2016]).”

18. Computer Science › Constraint Satisfaction and Optimization

    On the Solution-Space Geometry of Random Constraint Satisfaction Problems

    Achlioptas and Ricci‐Tersenghi · arXiv (Cornell University) · 2006

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    1. Unchecked“In particular, we prove that much before solutions disappear, they organize into an exponential number of clusters, each of which is relatively small and far apart from all other clusters.”
    2. Unchecked“Moreover, inside each cluster most variables are frozen, i.e., take only one value.”

19. Computer Science › Constraint Satisfaction and Optimization

    Algorithmic Barriers from Phase Transitions

    Achlioptas and Coja-Oghlan · Annual Symposium on Foundations of Computer Science · 2008

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    1. Unchecked“We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set.”
    2. Unchecked“We prove that a completely analogous phase transition also occurs both in random $k$-SAT and in random hypergraph 2-coloring.”