The solution space geometry of random linear equations

Random Struct Algorithms

2017

Self Cite

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.

“…The threshold for the appearance of a non‐empty k ‐core in

{scriptH}_{r} (n, c n)

(

(k, r) \neq (2, 2)

) is determined , given as below.

c_{r, k} = \inf_{μ > 0} \frac{normalμ}{r {true[e^{- μ} \sum_{i = k - 1}^{\infty} {normalμ}^{i} / i! true]}^{r - 1}} .

Define

c_{r}^{*} = c_{r, 2} .

The following theorem confirms that the clustering threshold for

X_{r} (n, c n)

c_{r}^{*}

.Theorem . For every fixed integer

r \geq 3

and real number

ϵ > 0

, if

c \leq c_{r}^{*} - ϵ

, then all solutions of

X_{r} (n, c n)

are

O (\log n)

‐connected; if

c \geq c_{r}^{*} + ϵ

then the solutions of

X_{r} (n, c n)

are partitioned into well‐connected well‐separated clusters: every cluster is

O

…”

Section: Resultsmentioning

confidence: 87%

c_{r}^{*}

, f ( n ) can be chosen as

C \log n

for some sufficiently large constant C > 0.…”

Section: Resultsmentioning

confidence: 99%

Inside the clustering window for random linear equations

Gao

Random Struct Algorithms

2017

Self Cite

“…The clusters of k-XOR-SAT are very well-understood, independently by [6,28] (see also [21]). We know the clustering threshold, which in this case is equal to the freezing threshold, and have a very good description of the clusters and the frozen variables.…”

Section: Minimal Rearrangementsmentioning

confidence: 95%

“…With probability at least 1 − e −f (n) , |D| = o(n). To just prove that all non-cyclic ∆-sets have size at least 2αn, we can use an approach that has been used in [47,15,6] to prove similar results: We would like to prove that the 2-core of every non-cyclic ∆-set has high edge-density. If we could do so, then a very fast and common argument based on subgraph densities in G n,M proves that every such subgraph must have linear size.…”

Section: Proofmentioning

confidence: 99%

See 1 more Smart Citation

The freezing threshold for k-colourings of a random graph

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

2012

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We rigorously determine the exact freezing threshold, r f k , for k-colourings of a random graph. We prove that for random graphs with density above r f k , almost every colouring is such that a linear number of variables are frozen, meaning that their colours cannot be changed by a sequence of alterations whereby we change the colours of o(n) vertices at a time, always obtaining another proper colouring. When the density is below r f k , then almost every colouring has at most o(n) frozen variables. This confirms hypotheses made using the non-rigorous cavity method.It has been hypothesized that the freezing threshold is the cause of the "algorithmic barrier", the long observed phenomenon that once the edge-density of a random graph passes 1 2 k ln k(1 + o k (1)), no algorithms are proven to find k-colourings, despite the fact that this density is only half the k-colourability threshold.We also show that r f k is the threshold of a strong form of reconstruction for k-colourings of the Galton-Watson tree, and of the graphical model. *

The stripping process can be slow: Part I

Gao

Random Struct Algorithms

2018

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Given an integer k, we consider the parallel k‐stripping process applied to a hypergraph H: removing all vertices with degree less than k in each iteration until reaching the k‐core of H. Take H as scriptHr(n,m): a random r‐uniform hypergraph on n vertices and m hyperedges with the uniform distribution. Fixing k,r≥2 with (k,r)≠(2,2), it has previously been proved that there is a constant cr,k such that for all m = cn with constant c≠cr,k, with high probability, the parallel k‐stripping process takes O(log⁡n) iterations. In this paper, we investigate the critical case when c=cr,k+o(1). We show that the number of iterations that the process takes can go up to some power of n, as long as c approaches cr,k sufficiently fast. A second result we show involves the depth of a non‐k‐core vertex v: the minimum number of steps required to delete v from scriptHr(n,m) where in each step one vertex with degree less than k is removed. We will prove lower and upper bounds on the maximum depth over all non‐k‐core vertices.