Mark Jerrum
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View article: Zero-free regions for the independence polynomial on restricted graph classes
Zero-free regions for the independence polynomial on restricted graph classes Open
Generalising the Heilman-Lieb Theorem from statistical physics, Chudnovsky and Seymour [J. Combin. Theory Ser. B, 97(3):350--357] showed that the univariate independence polynomial of any claw-free graph has all of its zeros on the negativ…
View article: Glauber dynamics for the hard-core model on bounded-degree $H$-free graphs
Glauber dynamics for the hard-core model on bounded-degree $H$-free graphs Open
The hard-core model has as its configurations the independent sets of some graph instance $G$ . The probability distribution on independent sets is controlled by a ‘fugacity’ $\lambda \gt 0$ , with higher $\lambda$ leading to denser config…
View article: Rapid mixing of the flip chain over non-crossing spanning trees
Rapid mixing of the flip chain over non-crossing spanning trees Open
We show that the flip chain for non-crossing spanning trees of $n+1$ points in convex position mixes in time $O(n^8\log n)$. We use connections between Fuss-Catalan structures to construct a comparison argument with a chain similar to Wils…
View article: Perfect sampling of $q$-spin systems on $\mathbb{Z}^{2}$ via weak spatial mixing
Perfect sampling of $q$-spin systems on $\mathbb{Z}^{2}$ via weak spatial mixing Open
We present a perfect marginal sampler of the unique Gibbs measure of a spin system on \mathbb{Z}^{2} . The algorithm is an adaptation of a previous “lazy depth-first” approach by the authors, but relaxes the requirement of strong spatial m…
View article: Glauber dynamics for the hard-core model on bounded-degree $H$-free graphs
Glauber dynamics for the hard-core model on bounded-degree $H$-free graphs Open
The hard-core model has as its configurations the independent sets of some graph instance $G$. The probability distribution on independent sets is controlled by a `fugacity' $λ>0$, with higher $λ$ leading to denser configurations. We inves…
View article: Fundamentals of partial rejection sampling
Fundamentals of partial rejection sampling Open
Partial Rejection Sampling is an algorithmic approach to obtaining a perfect\nsample from a specified distribution. The objects to be sampled are assumed to\nbe represented by a number of random variables. In contrast to classical\nrejecti…
View article: A simple polynomial-time approximation algorithm for the total variation distance between two product distributions
A simple polynomial-time approximation algorithm for the total variation distance between two product distributions Open
We give a simple polynomial-time approximation algorithm for the total variation distance between two product distributions.
View article: Perfect Sampling of $q$-Spin Systems on $\mathbb Z^2$ via Weak Spatial Mixing
Perfect Sampling of $q$-Spin Systems on $\mathbb Z^2$ via Weak Spatial Mixing Open
We present a perfect marginal sampler of the unique Gibbs measure of a spin system on $\mathbb Z^2$. The algorithm is an adaptation of a previous `lazy depth-first' approach by the authors, but relaxes the requirement of strong spatial mix…
View article: A simple polynomial-time approximation algorithm for the total variation distance between two product distributions
A simple polynomial-time approximation algorithm for the total variation distance between two product distributions Open
We give a simple polynomial-time approximation algorithm for the total variation distance between two product distributions.
View article: A simple polynomial-time approximation algorithm for the total variation distance between two product distributions
A simple polynomial-time approximation algorithm for the total variation distance between two product distributions Open
We give a simple polynomial-time approximation algorithm for the total variation distance between two product distributions.
View article: Counting Vertices of Integral Polytopes Defined by Facets
Counting Vertices of Integral Polytopes Defined by Facets Open
We present a number of complexity results concerning the problem of counting vertices of an integral polytope defined by a system of linear inequalities. The focus is on polytopes with small integer vertices, particularly 0/1 polytopes and…
View article: Counting weighted independent sets beyond the permanent
Counting weighted independent sets beyond the permanent Open
Jerrum, Sinclair, and Vigoda [J. ACM, 51 (2004), pp. 671--697] showed that the permanent of any square matrix can be estimated in polynomial time. This computation can be viewed as approximating the partition function of edge-weighted matc…
View article: Perfect Sampling in Infinite Spin Systems via Strong Spatial Mixing
Perfect Sampling in Infinite Spin Systems via Strong Spatial Mixing Open
We present a simple algorithm that perfectly samples configurations from the unique Gibbs measure of a spin system on a potentially infinite graph $G$. The sampling algorithm assumes strong spatial mixing together with subexponential growt…
View article: Fundamentals of Partial Rejection Sampling
Fundamentals of Partial Rejection Sampling Open
Partial Rejection Sampling is an algorithmic approach to obtaining a perfect sample from a specified distribution. The objects to be sampled are assumed to be represented by a number of random variables. In contrast to classical rejection …
View article: Counting vertices of integral polytopes defined by facets
Counting vertices of integral polytopes defined by facets Open
We present a number of complexity results concerning the problem of counting vertices of an integral polytope defined by a system of linear inequalities. The focus is on polytopes with small integer vertices, particularly 0/1 polytopes and…
View article: Counting vertices of integer polytopes defined by facets.
Counting vertices of integer polytopes defined by facets. Open
We present a number of complexity results concerning the problem of counting vertices of an integral polytope defined by a system of linear inequalities. The focus is on polytopes with small integer vertices, particularly 0/1 polytopes and…
View article: Polynomial-time approximation algorithms for the antiferromagnetic Ising model on line graphs
Polynomial-time approximation algorithms for the antiferromagnetic Ising model on line graphs Open
We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the ‘winding’ technology devised by McQuil…
View article: Perfect simulation of the hard disks model by partial rejection sampling
Perfect simulation of the hard disks model by partial rejection sampling Open
We present a perfect simulation of the hard disks model via the partial rejection sampling method. Provided the density of disks is not too high, the method produces exact samples in O(log n) rounds, where n is the expected number of disks…
View article: The Size of the Giant Joint Component in a Binomial Random Double Graph
The Size of the Giant Joint Component in a Binomial Random Double Graph Open
We study the joint components in a random 'double graph' that is obtained by superposing red and blue binomial random graphs on $n$~vertices. A joint component is a maximal set of vertices that supports both a red and a blue spanning tree.…
View article: Counting Weighted Independent Sets beyond the Permanent
Counting Weighted Independent Sets beyond the Permanent Open
Jerrum, Sinclair and Vigoda (2004) showed that the permanent of any square matrix can be estimated in polynomial time. This computation can be viewed as approximating the partition function of edge-weighted matchings in a bipartite graph. …
View article: Approximately counting bases of bicircular matroids
Approximately counting bases of bicircular matroids Open
We give a fully polynomial-time randomized approximation scheme (FPRAS) for the number of bases in bicircular matroids. This is a natural class of matroids for which counting bases exactly is # P -hard and yet approximate counting can be d…
View article: Random Walks on Small World Networks
Random Walks on Small World Networks Open
We study the mixing time of random walks on small-world networks modelled as follows: starting with the 2-dimensional periodic grid, each pair of vertices {u,v} with distance d> 1 is added as a “long-range” edge with probability proportion…
View article: Approximating Pairwise Correlations in the Ising Model
Approximating Pairwise Correlations in the Ising Model Open
In the Ising model, we consider the problem of estimating the covariance of the spins at two specified vertices. In the ferromagnetic case, it is easy to obtain an additive approximation to this covariance by repeatedly sampling from the r…
View article: Uniform Sampling Through the Lovász Local Lemma
Uniform Sampling Through the Lovász Local Lemma Open
We propose a new algorithmic framework, called partial rejection sampling , to draw samples exactly from a product distribution, conditioned on none of a number of bad events occurring. Our framework builds new connections between the vari…
View article: Random cluster dynamics for the Ising model is rapidly mixing
Random cluster dynamics for the Ising model is rapidly mixing Open
We show that the mixing time of Glauber (single edge update) dynamics for the random cluster model at $q=2$ on an arbitrary $n$-vertex graph is bounded by a polynomial in $n$. As a consequence, the Swendsen–Wang algorithm for the ferromagn…
View article: Perfect Simulation of the Hard Disks Model by Partial Rejection Sampling
Perfect Simulation of the Hard Disks Model by Partial Rejection Sampling Open
We present a perfect simulation of the hard disks model via the partial rejection sampling method. Provided the density of disks is not too high, the method produces exact samples in O(log n) rounds, where n is the expected number of disks…
View article: Computational Counting (Dagstuhl Seminar 17341)
Computational Counting (Dagstuhl Seminar 17341) Open
This report documents the program and the outcomes of Dagstuhl Seminar 17341 "Computational Counting". The seminar was held from 20th to 25th August 2017, at Schloss Dagstuhl -- Leibnitz Center for Informatics. A total of 43 researchers fr…
View article: A simple FPRAS for bi-directed reachability.
A simple FPRAS for bi-directed reachability. Open
Gorodezky and Pak (Random Struct. Algorithms, 2014) introduced a cluster-popping algorithm for sampling root-connected subgraphs in a directed graph, and conjectured that it runs in expected polynomial time on bi-directed graphs. We confir…
View article: A polynomial-time approximation algorithm for all-terminal network reliability
A polynomial-time approximation algorithm for all-terminal network reliability Open
We give a fully polynomial-time randomized approximation scheme (FPRAS) for the all-terminal network reliability problem, which is to determine the probability that, in a undirected graph, assuming each edge fails independently, the remain…