The Satisfiability Threshold For Random Linear Equations

Ayre, Peter J.; Coja-Oghlan, Amin; Gao, Pu; Müller, Noela

doi:10.1007/s00493-019-3897-3

Cited by 15 publications

(48 citation statements)

References 53 publications

(102 reference statements)

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“…Motivated by the minimum spanning tree problem in weighted random graphs, Cooper, Frieze and Pegden [14] studied the rank of the random matrix with degree distributions k = k ≥ 3 fixed and d ∼ Po(d) over the field F 2 . The same rank formula was obtained independently in [4]. Extending the results from [4,14], Theorem 1.1 shows that the rank of the random matrix with these degrees over any field F is given by…”

Section: Introductionsupporting

confidence: 64%

“…The same rank formula was obtained independently in [4]. Extending the results from [4,14], Theorem 1.1 shows that the rank of the random matrix with these degrees over any field F is given by…”

Section: Introductionsupporting

confidence: 64%

“…Ayre, Coja-Oghlan, Gao and Müller [4] proposed a different strategy to cope with sparse random matrices with precisely k non-zero entries per row over general finite fields (the model from Example 1.3). Instead of performing a moment calculation, [4] relies on a coupling argument inspired by the Aizenman-Sims-Starr scheme from mathematical physics [1]. The basic idea is to set up a coupling of a random matrix with n columns and a slightly larger random matrix with n + 1 columns and to estimate the expected difference of their nullities.…”

Section: Discussion and Related Workmentioning

confidence: 99%

“…A key lemma from [4] deals with dependencies among bounded numbers of entries of σ. Specifically, the following definition introduces a small perturbation, applicable to any matrix, which, we will show, mostly eliminates dependencies among small numbers of coordinates.…”

Section: 2mentioning

confidence: 99%

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The rank of sparse random matrices

Coja-Oghlan¹,

Ergür²,

Gao³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Generalising prior work on the rank of random matrices over finite fields [Coja-Oghlan and Gao 2018], we determine the rank of a random matrix A with prescribed numbers of non-zero entries in each row and column over any field F. The rank formula turns out to be independent of both the field and the distribution of the non-zero matrix entries. The proofs are based on a blend of algebraic and probabilistic methods inspired by ideas from mathematical physics.MSC: 05C80, 60B20, 94B05Amin Coja-Oghlan's research received support under DFG CO 646/3. Alperen A. Ergür partially funded by Einstein Foundation, Berlin.

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Section: Introductionsupporting

confidence: 64%

Section: Introductionsupporting

confidence: 64%

Section: Discussion and Related Workmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

See 2 more Smart Citations

The rank of sparse random matrices

Coja-Oghlan¹,

Ergür²,

Gao³

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Let c * k be the value of c for which the 2-core has asymptotically the same number of vertices and edges. More precisely, we use (2) and 3to define c * k by c * k := min c c k :…”

Section: Matrix Rankmentioning

confidence: 99%

On the Rank of a Random Binary Matrix

Cooper

Frieze

Pegden

2019

Electron. J. Combin.

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We study the rank of a random $n \times m$ matrix $\mathbf{A}_{n,m;k}$ with entries from $GF(2)$, and exactly $k$ unit entries in each column, the other entries being zero. The columns are chosen independently and uniformly at random from the set of all ${n \choose k}$ such columns. We obtain an asymptotically correct estimate for the rank as a function of the number of columns $m$ in terms of $c,n,k$, and where $m=cn/k$. The matrix $\mathbf{A}_{n,m;k}$ forms the vertex-edge incidence matrix of a $k$-uniform random hypergraph $H$. The rank of $\mathbf{A}_{n,m;k}$ can be expressed as follows. Let $|C_2|$ be the number of vertices of the 2-core of $H$, and $|E(C_2)|$ the number of edges. Let $m^*$ be the value of $m$ for which $|C_2|= |E(C_2)|$. Then w.h.p. for $m<m^*$ the rank of $\mathbf{A}_{n,m;k}$ is asymptotic to $m$, and for $m \ge m^*$ the rank is asymptotic to $m-|E(C_2)|+|C_2|$. In addition, assign i.i.d. $U[0,1]$ weights $X_i, i \in {1,2,...m}$ to the columns, and define the weight of a set of columns $S$ as $X(S)=\sum_{j \in S} X_j$. Define a basis as a set of $n-𝟙 (k\text{ even})$ linearly independent columns. We obtain an asymptotically correct estimate for the minimum weight basis. This generalises the well-known result of Frieze [On the value of a random minimum spanning tree problem, Discrete Applied Mathematics, (1985)] that, for $k=2$, the expected length of a minimum weight spanning tree tends to $\zeta(3)\sim 1.202$.

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The number of satisfying assignments of random 2‐SAT formulas

Achlioptas

Coja-Oghlan

Hahn‐Klimroth

et al. 2021

Random Struct Algorithms

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We show that throughout the satisfiable phase the normalized number of satisfying assignments of a random 2-SAT formula converges in probability to an expression predicted by the cavity method from statistical physics. 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. KEYWORDS 2-SAT, Belief Propagation, satisfiability problem 1 INTRODUCTION Background and motivationThe random 2-SAT problem was the first random constraint satisfaction problem whose satisfiability threshold could be pinpointed precisely, an accomplishment attained independently by Chvátal and Reed [15] and Goerdt [31] in 1992. The proofs evince the link between the 2-SAT threshold and the percolation phase transition of a random digraph. This connection subsequently enabled Bollobás, Borgs, Chayes, Kim, and Wilson [12] to identify the size of the scaling window, which matches thatThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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The Satisfiability Threshold For Random Linear Equations

Cited by 15 publications

References 53 publications

The rank of sparse random matrices

The rank of sparse random matrices

On the Rank of a Random Binary Matrix

The number of satisfying assignments of random 2‐SAT formulas

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