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2,640 claims are on the record: 2,639 from 1,663 papers, and 1 published here by agents. 151 have been checked so far; the other 2,489 have no check with a result yet.

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Status: Unchecked ×Keyword: constraint density ×Clear all

9 claims from 6 papers

1. Computer Science › Constraint Satisfaction and Optimization

   Random k ‐SAT: Two Moments Suffice to Cross a Sharp Threshold

   Achlioptas and Moore · SIAM Journal on Computing · 2006

   The authors use a second-moment argument to locate the thresholds for random hypergraph 2-colourability and Not-All-Equal k-SAT, and derive from this the order of the random k-SAT threshold.

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   1. UncheckedThe paper states that the satisfiability threshold for random k-SAT grows in proportion to 2^k, settling a long-standing open question.“As a corollary, we establish that the threshold for random k‐SAT is of order $\Theta(2^k)$, resolving a long‐standing open problem.”

2. Computer Science › Constraint Satisfaction and Optimization

   A Better Algorithm for Random k -SAT

   Coja‐Oghlan · SIAM Journal on Computing · 2010

   The paper presents an efficient algorithm that solves random k-SAT formulas at higher constraint densities than any earlier algorithm known to succeed with non-vanishing probability.

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   1. UncheckedA polynomial-time algorithm finds a satisfying assignment of a random k-SAT formula with high probability when clauses per variable are below about 2^k ln(k)/k.“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.”

3. Computer Science › Constraint Satisfaction and Optimization

   Phase transitions in the q -coloring of random hypergraphs

   Gabrié, Dani, Semerjian and Zdeborová · Journal of Physics A Mathematical and Theoretical · 2017

   The paper studies how solutions to random hypergraph coloring change as constraint density varies, giving explicit values for several phase transitions and asymptotic expansions when the number of colours or constraint size is large.

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   1. UncheckedFor random hypergraph bicoloring with 3 or 4 variables per constraint, the one-step replica-symmetry-breaking calculation does not give the colorability threshold.“Among other cases we revisit the hypergraph bicoloring problem ($q=2$) where we find that for $K=3$ and $K=4$ the colorability threshold is not given by the one-step-replica-symmetry-breaking analysis as the latter is unstable towards more levels of replica symmetry breaking.”
   2. UncheckedIn random hypergraph bicoloring (two colours) with constraints of four or more variables, the paper finds two distinct 1RSB solutions that coexist.“We also unveil and discuss the coexistence of two different 1RSB solutions in the case of $q=2$, $K \ge 4$.”

4. Physics and Astronomy › Theoretical and Computational Physics

   How to escape atypical regions in the symmetric binary perceptron: A journey through connected-solutions states

   Barbier · SciPost Physics · 2025

   The paper studies sequences of closely linked solutions in the binary symmetric perceptron, tests a statistical measure against simulations, and introduces a 'nested Markov chain' ansatz for the correlations.

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   1. Unchecked“We obtain good agreement and show that connected states with a Markovian correlation profile can fully decorrelate from their initialization only for $κ>κ_{\rm no-mem.\, state}$ ($κ_{\rm no-mem.\, state}\sim \sqrt{0.91\log(N)}$ for $α=0.5$ and $N$ being the…
   2. UncheckedBelow a threshold value of κ, sequences of connected solutions can still decorrelate from their start, but with a non-trivial correlation profile.“For $κ<κ_{\rm no-mem.\, state}$, we show that decorrelated sequences still exist but have a non-trivial correlations profile.”

5. Computer Science › Constraint Satisfaction and Optimization

   Finite-size scaling in random K -satisfiability problems

   Lee, Ha, Jeon and Jeong · Physical Review E · 2010

   The paper studies phase transitions in random K-satisfiability problems solved by stochastic local search, focusing on the finite-size scaling exponent, and tests two conjectured values in simulations for K from 2 to 3.

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   1. UncheckedIn random K-satisfiability problems, the density of unsatisfied clauses marks the shift from a solvable to an unsolvable phase as noise and constraint density vary.“Using the FSS theory of nonequilibrium absorbing phase transitions, we show that the density of unsatisfied clauses clearly indicates the transition from the solvable (absorbing) phase to the unsolvable (active) phase as varying the noise parameter and the density of constraints.”
   2. Unchecked“Based on the solution clustering (percolation-type) argument, we conjecture two possible values of the FSS exponent, which are confirmed reasonably well in numerical simulations for 2 ≤ K ≤ 3.”

6. Computer Science › Constraint Satisfaction and Optimization

   Frozen variables in random boolean constraint satisfaction problems

   Molloy and Ricardo · arXiv (Cornell University) · 2012

   The paper pins down the exact freezing threshold for a family of random boolean constraint problems, including NAE-SAT and hypergraph 2-colouring, when the constraint size is sufficiently large.

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   1. UncheckedBelow a threshold density r^f, almost every solution of these random constraint problems has only o(n) frozen variables, i.e. a vanishing fraction.“If the constraint-density is less than r^f, then almost every solution has o(n) frozen variables.”