Ewan Davies
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View article: Safety Analysis in the NGAC Model
Safety Analysis in the NGAC Model Open
We study the safety problem for the next-generation access control (NGAC) model. We show that under mild assumptions it is coNP-complete, and under further realistic assumptions we give an algorithm for the safety problem that significantl…
View article: On expectations and variances in the hard-core model on bounded degree graphs
On expectations and variances in the hard-core model on bounded degree graphs Open
We extend the study of the occupancy fraction of the hard-core model in two novel directions. One direction gives a tight lower bound in terms of individual vertex degrees, extending work of Sah, Sawhney, Stoner and Zhao which bounds the p…
View article: Local Weak Degeneracy of Planar Graphs
Local Weak Degeneracy of Planar Graphs Open
Thomassen showed that planar graphs are 5-list-colourable, and that planar graphs of girth at least five are 3-list-colourable. An easy degeneracy argument shows that planar graphs of girth at least four are 4-list-colourable. In 2022, Pos…
View article: The hard-core model in graph theory
The hard-core model in graph theory Open
An independent set may not contain both a vertex and one of its neighbours. This basic fact makes the uniform distribution over independent sets rather special. We consider the hard-core model, an essential generalization of the uniform di…
View article: Sampling List Packings
Sampling List Packings Open
We initiate the study of approximately counting the number of list packings of a graph. The analogous problem for usual vertex coloring and list coloring has attracted substantial attention. For list packing the setup is similar, but we se…
View article: On the occupancy fraction of the antiferromagnetic Ising model
On the occupancy fraction of the antiferromagnetic Ising model Open
We study the maximum and minimum occupancy fraction of the antiferromagnetic Ising model in regular graphs. The minimizing problem is known to determine a computational threshold in the complexity of approximately sampling from the Ising m…
View article: Algorithms for the ferromagnetic Potts model on expanders
Algorithms for the ferromagnetic Potts model on expanders Open
We give algorithms for approximating the partition function of the ferromagnetic $q$ -color Potts model on graphs of maximum degree $d$ . Our primary contribution is a fully polynomial-time approximation scheme for $d$ -regular graphs with…
View article: A robust Corrádi–Hajnal theorem
A robust Corrádi–Hajnal theorem Open
For a graph and , we denote by the random sparsification of obtained by keeping each edge of independently, with probability . We show that there exists a such that if and is an ‐vertex graph with and , then with high probability contains …
View article: A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs
A Spectral Approach to Approximately Counting Independent Sets in Dense Bipartite Graphs Open
We give a randomized algorithm that approximates the number of independent sets in a dense, regular bipartite graph - in the language of approximate counting, we give an FPRAS for #BIS on the class of dense, regular bipartite graphs. Effic…
View article: Detecting trends and shocks in terrorist activities
Detecting trends and shocks in terrorist activities Open
Although there are some techniques for dealing with sparse and concentrated discrete data, standard time-series analyses appear ill-suited to understanding the temporal patterns of terrorist attacks due to the sparsity of the events. This …
View article: Packing list‐colorings
Packing list‐colorings Open
List coloring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list‐coloring, we seek many in parallel. Our explorations have uncovered a potentiall…
View article: List packing number of bounded degree graphs
List packing number of bounded degree graphs Open
We investigate the list packing number of a graph, the least $k$ such that there are always $k$ disjoint proper list-colourings whenever we have lists all of size $k$ associated to the vertices. We are curious how the behaviour of the list…
View article: A robust Corrádi--Hajnal Theorem
A robust Corrádi--Hajnal Theorem Open
For a graph $G$ and $p\in[0,1]$, we denote by $G_p$ the random sparsification of $G$ obtained by keeping each edge of $G$ independently, with probability $p$. We show that there exists a $C>0$ such that if $p\geq C(\log n)^{1/3}n^{-2/3}$ a…
View article: The $\chi$-Ramsey Problem for Triangle-Free Graphs
The $\chi$-Ramsey Problem for Triangle-Free Graphs Open
In 1967, Erd\H{o}s asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erd\H{o}s and Hajnal together with Shearer's classical upper bound for the off-diagonal Ramsey number $R(3,…
View article: Algorithms for the ferromagnetic Potts model on expanders
Algorithms for the ferromagnetic Potts model on expanders Open
We give algorithms for approximating the partition function of the ferromagnetic $q$-color Potts model on graphs of maximum degree $d$. Our primary contribution is a fully polynomial-time approximation scheme for $d$-regular graphs with an…
View article: Computational thresholds for the fixed-magnetization Ising model
Computational thresholds for the fixed-magnetization Ising model Open
The ferromagnetic Ising model is a model of a magnetic material and a central topic in statistical physics. It also plays a starring role in the algorithmic study of approximate counting: approximating the partition function of the ferroma…
View article: Packing list-colourings
Packing list-colourings Open
List colouring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-colouring, we seek many in parallel. Our explorations have uncovered a potentia…
View article: On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs
On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs Open
For a graph G=(V,E) , k\in \mathbb{N} , and complex numbers w=(w_e)_{e\in E} the partition function of the multivariate Potts model is defined as \mathbf{Z}(G;k,w):=\sum_{\phi\colon V\to [k]} \prod_{\substack{e=uv\in E \\ \phi(u)=\phi(v)}}…
View article: The $χ$-Ramsey problem for triangle-free graphs
The $χ$-Ramsey problem for triangle-free graphs Open
In 1967, Erdős asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erdős and Hajnal together with Shearer's classical upper bound for the off-diagonal Ramsey number $R(3, t)$ sho…
View article: Occupancy fraction, fractional colouring, and triangle fraction
Occupancy fraction, fractional colouring, and triangle fraction Open
Given $\varepsilon>0$, there exists $f_0$ such that, if $f_0 \le f \le Δ^2+1$, then for any graph $G$ on $n$ vertices of maximum degree $Δ$ in which the neighbourhood of every vertex in $G$ spans at most $Δ^2/f$ edges, (i) an independent s…
View article: Approximately counting independent sets of a given size in bounded-degree graphs
Approximately counting independent sets of a given size in bounded-degree graphs Open
We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density $α_c(Δ)$ and provide (i) for $α< α_c(Δ)$ randomized polynom…
View article: Approximately Counting Independent Sets of a Given Size in Bounded-Degree Graphs
Approximately Counting Independent Sets of a Given Size in Bounded-Degree Graphs Open
We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density α_c(Δ) and provide (i) for α < α_c(Δ) randomized polynomial…
View article: An Approximate Blow-up Lemma for Sparse Hypergraphs
An Approximate Blow-up Lemma for Sparse Hypergraphs Open
We obtain an approximate sparse hypergraph version of the blow-up lemma, showing that partite hypergraphs with sufficient regularity of small subgraph counts behave as if they were complete partite for the purpose of embedding bounded degr…
View article: Coloring triangle‐free graphs with local list sizes
Coloring triangle‐free graphs with local list sizes Open
We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow‐up work of Bernshteyn) on the (list) chromatic number of triangle‐free graphs. In both our results, we permit the amount of color made av…
View article: An algorithmic framework for colouring locally sparse graphs
An algorithmic framework for colouring locally sparse graphs Open
We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$, a randomised polynomial-ti…
View article: Graph structure via local occupancy
Graph structure via local occupancy Open
The first author together with Jenssen, Perkins and Roberts (2017) recently showed how local properties of the hard-core model on triangle-free graphs guarantee the existence of large independent sets, of size matching the best-known asymp…
View article: Efficient algorithms for the Potts model on small-set expanders
Efficient algorithms for the Potts model on small-set expanders Open
An emerging trend in approximate counting is to show that certain `low-temperature' problems are easy on typical instances, despite worst-case hardness results. For the class of regular graphs one usually shows that expansion can be exploi…
View article: Statistical physics approaches to Unique Games
Statistical physics approaches to Unique Games Open
We show how two techniques from statistical physics can be adapted to solve a variant of the notorious Unique Games problem, potentially opening new avenues towards the Unique Games Conjecture. The variant, which we call Count Unique Games…