Corpus ID: 119600241

The Satisfiability Threshold for $k$-XORSAT, using an alternative proof

  title={The Satisfiability Threshold for \$k\$-XORSAT, using an alternative proof},
  author={Boris G. Pittel and Gregory B. Sorkin},
  journal={arXiv: Combinatorics},
We consider "unconstrained" random $k$-XORSAT, which is a uniformly random system of $m$ linear non-homogeneous equations in $\mathbb{F}_2$ over $n$ variables, each equation containing $k \ge 3$ variables, and also consider a "constrained" model where every variable appears in at least two equations. Dubois and Mandler proved that $m/n=1$ is a sharp threshold for satisfiability of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform hypergraph to extend this result to find the… Expand
5 Citations

Figures from this paper

The Satisfiability Threshold for k-XORSAT
  • B. Pittel, G. Sorkin
  • Mathematics, Computer Science
  • Combinatorics, Probability and Computing
  • 2015
It is shown that m/n = 1 remains a sharp threshold for satisfiability of constrained k-XORSAT for every k ⩾ 3, and standard results on the 2-core of a random k-uniform hypergraph are used to extend this result to find the threshold for unconstrained k- XORSAT. Expand
Satisfiability threshold for random regular NAE-SAT
This work considers the random regular k-nae-sat problem with n variables each appearing in exactly d clauses and establishes explicitly the satisfiability threshold d* ∈ d*(k), which is the first result to locate the exact Satisfiability threshold in a random constraint satisfaction problem exhibiting the condensation phenomenon. Expand
On the phase transition in random simplicial complexes
It is well-known that the $G(n,p)$ model of random graphs undergoes a dramatic change around $p=\frac 1n$. It is here that the random graph is, almost surely, no longer a forest, and here it firstExpand
Concentration of the number of solutions of random planted CSPs and Goldreich's one-way candidates
It is shown that the logarithm of the number of solutions of a random planted k-SAT formula concentrates around a deterministic n-independent threshold and is extended to a more general class of random planted CSPs. Expand
Conditional Random Fields, Planted Constraint Satisfaction and Entropy Concentration
This paper studies a class of probabilistic models on graphs, where edge variables depend on incident node variables through a fixed probability kernel, and shows that under mild assumptions on the kernel, the conditional entropy of the node variables given the edge variables concentrates around a deterministic threshold. Expand


Rank deficiency in sparse random GF$[2]$ matrices
Let $M$ be a random $m \times n$ matrix with binary entries and i.i.d. rows. The weight (i.e., number of ones) of a row has a specified probability distribution, with the row chosen uniformly atExpand
How Frequently is a System of 2-Linear Boolean Equations Solvable?
The Boolean system with $b_e\equiv 1$ is solvable iff the underlying graph is $2$-colorable, and it is shown that probability of $2-colorability is $\lesssim 2^{-1/4}e^{1/8}c(\lambda)n^{- 1/12}$. Expand
Random 2-XORSAT at the Satisfiability Threshold
This study relies on the symbolic method and analytical tools coming from generating function theory which enable it to describe the evolution of n1/12 p(n, n/2(1 + µn-1/3)) as a function of µ. Expand
The scaling window of the 2-SAT transition
Using this order parameter, it is proved that the 2-SAT phase transition is continuous with an order parameter critical exponent of 1 and the values of two other critical exponents are determined, showing that the exponents of 2- SAT are identical to those of the random graph. Expand
A Threshold for Unsatisfiability
Due to the close relationship between satisfiability of formulas in 2-CNF and graph theoretic properties it is not surprising that the proof uses techniques from the theory of random graphs, in particular [12]. Expand
Random MAX SAT, random MAX CUT, and their phase transitions
It is possible that optimization problems may prove easier to analyze than their decision analogs, and may help to elucidate them, and it is proved that random 2-SAT formulas with clause/variable density less than 1 are almost always satisfiable, and the "scaling window" is in the density range 1 ± Θ (n-1/3). Expand
Smooth and sharp thresholds for random k-XOR-CNF satisfiability
For k ≥ 3 the authors show the existence of a sharp threshold for the satisfiability of a random k -XOR-CNF formula, whereas there are smooth thresholds for k=1 and k=2 . Expand
Mick gets some (the odds are on his side) (satisfiability)
  • V. Chvátal, B. Reed
  • Mathematics, Computer Science
  • Proceedings., 33rd Annual Symposium on Foundations of Computer Science
  • 1992
The authors present a linear-time algorithm that satisfies F with probability 1-o(1) whenever c<(0.25)2/sup k//k and establish a threshold for 2-SAT: if k = 2 then F is satisfiable with probability1-o (1) Whenever c<1 and unsatisfiable with probabilities 1-O(1), whenever c>1. Expand
Tight Thresholds for Cuckoo Hashing via XORSAT
The question of tight thresholds for offline cuckoo hashing is settled by finding the thresholds for all values of k > 2, which are in fact the same as the previously known thresholds for the random k-XORSAT problem. Expand
Poisson Cloning Model for Random Graphs
In the random graph G(n, p) with pn bounded, the degrees of the vertices are almost i.i.d Poisson random variables with mean λ := p(n − 1). Motivated by this fact, we introduce the Poisson cloningExpand