Deterministic counting of graph colourings using sequences of subgraphs

In this paper we propose a deterministic algorithm for approximately counting the $k$-colourings of sparse random graphs $G(n,d/n)$. In particular, our algorithm computes in polynomial time a $(1\pm n^{-\Omega(1)})$approximation of the logarithm of the number of $k$-colourings of $G(n,d/n)$ for $k\geq (2+\epsilon) d$ with high probability over the graph instances. Our algorithm is related to the algorithms of A. Bandyopadhyay et al. in SODA '06, and A. Montanari et al. in SODA '06, i.e. it uses {\em spatial correlation decay} to compute {\em deterministically} marginals of {\em Gibbs distribution}. We develop a scheme whose accuracy depends on {\em non-reconstruction} of the colourings of $G(n,d/n)$, rather than {\em uniqueness} that are required in previous works. This leaves open the possibility for our schema to be sufficiently accurate even for $k<d$. The set up for establishing correlation decay is as follows: Given $G(n,d/n)$, we alter the graph structure in some specific region $\Lambda$ of the graph by deleting edges between vertices of $\Lambda$. Then we show that the effect of this change on the marginals of Gibbs distribution, diminishes as we move away from $\Lambda$. Our approach is novel and suggests a new context for the study of deterministic counting algorithms

Reconstruction/Non-reconstruction Thresholds for Colourings of General Galton-Watson Trees

The broadcasting models on trees arise in many contexts such as discrete mathematics, biology statistical physics and cs. In this work, we consider the colouring model. A basic question here is whether the root's assignment affects the distribution of the colourings at the vertices at distance h from the root. This is the so-called "reconstruction problem". For a d-ary tree it is well known that d/ln (d) is the reconstruction threshold. That is, for k=(1+eps)d/ln(d) we have non-reconstruction while for k=(1-eps)d/ln(d) we have. Here, we consider the largely unstudied case where the underlying tree is chosen according to a predefined distribution. In particular, our focus is on the well-known Galton-Watson trees. This model arises naturally in many contexts, e.g. the theory of spin-glasses and its applications on random Constraint Satisfaction Problems (rCSP). The aforementioned study focuses on Galton-Watson trees with offspring distribution B(n,d/n), i.e. the binomial with parameters n and d/n, where d is fixed. Here we consider a broader version of the problem, as we assume general offspring distribution, which includes B(n,d/n) as a special case. Our approach relates the corresponding bounds for (non)reconstruction to certain concentration properties of the offspring distribution. This allows to derive reconstruction thresholds for a very wide family of offspring distributions, which includes B(n,d/n). A very interesting corollary is that for distributions with expected offspring d, we get reconstruction threshold d/ln(d) under weaker concentration conditions than what we have in B(n,d/n). Furthermore, our reconstruction threshold for the random colorings of Galton-Watson with offspring B(n,d/n), implies the reconstruction threshold for the random colourings of G(n,d/n)

Intermodal passenger transport and destination competitiveness in Greece

Effective transport is impeded by a number of caveats, including problems of accessibility to the destination, poor infrastructure, social, and environmental issues. In this context, the implementation of intermodal solutions is essential to meet customer demand, resolve problems of transport supply, and enhance destination competitiveness. Based on a suitable theoretical framework, this paper examines the attitude of Greek passengers towards intermodal transport and their willingness-to-pay more to be provided with such a seamless service to allow for (partial at least) cost recovery of the related transport infrastructure. The findings suggest that there are many respondents who would actually pay more to be provided with a door-to-door intermodal travel experience; answers are highly dependent on their place of residence

Local convergence of random graph colorings

Let $G=G(n,m)$ be a random graph whose average degree $d=2m/n$ is below the $k$-colorability threshold. If we sample a $k$-coloring $\sigma$ of $G$ uniformly at random, what can we say about the correlations between the colors assigned to vertices that are far apart? According to a prediction from statistical physics, for average degrees below the so-called {\em condensation threshold} $d_c(k)$, the colors assigned to far away vertices are asymptotically independent [Krzakala et al.: Proc. National Academy of Sciences 2007]. We prove this conjecture for $k$ exceeding a certain constant $k_0$. More generally, we investigate the joint distribution of the $k$-colorings that $\sigma$ induces locally on the bounded-depth neighborhoods of any fixed number of vertices. In addition, we point out an implication on the reconstruction problem

MCMC sampling colourings and independent sets of G(n,d/n) near the uniqueness threshold

Sampling from the Gibbs distribution is a well studied problem in computer science as well as in statistical physics. In this work we focus on the k-colouring model and the hard-core model with fugacity \lambda when the underlying graph is an instance of Erdos-Renyi random graph G(n,p), where p=d/n and d is fixed. We use the Markov Chain Monte Carlo method for sampling from the aforementioned distributions. In particular, we consider Glauber (block) dynamics. We show a dramatic improvement on the bounds for rapid mixing in terms of the number of colours and the fugacity for the corresponding models. For both models the bounds we get are only within small constant factors from the conjectured ones by the statistical physicists. We use Path Coupling to show rapid mixing. For k and \lambda in the range of our interest the technical challenge is to cope with the high degree vertices, i.e. vertices of degree much larger than the expected degree d. The usual approach to this problem is to consider block updates rather than single vertex updates for the Markov chain. Taking appropriately defined blocks the effect of high degree vertices diminishes. However devising such a block construction is a non trivial task. We develop for a first time a weighting schema for the paths of the underlying graph. Only, vertices which belong to "light" paths can be placed at the boundaries of the blocks. The tree-like local structure of G(n,d/n) allows the construction of simple structured blocks

A simple algorithm for sampling colourings of G(N,D/N) up to Gibbs Uniqueness threshold

Approximate random $k$-colouring of a graph G is a well studied problem in computer science and statistical physics. It amounts to constructing a k-colouring of G which is distributed close to Gibbs distribution in polynomial time. Here, we deal with the problem when the underlying graph is an instance of Erdos-Renyi random graph G(n,d/n), where d is a sufficiently large constant. We propose a novel efficient algorithm for approximate random k-colouring G(n,d/n) for any k>(1+\epsilon)d. To be more specific, with probability at least 1-n^{-\Omega(1)} over the input instances G(n,d/n) and for k>(1+\epsilon)d, the algorithm returns a k-colouring which is distributed within total variation distance n^{-\Omega(1)} from the Gibbs distribution of the input graph instance. The algorithm we propose is neither a MCMC one nor inspired by the message passing algorithms proposed by statistical physicists. Roughly the idea is as follows: Initially we remove sufficiently many edges of the input graph. This results in a ``simple graph" which can be $k$-coloured randomly efficiently. The algorithm colours randomly this simple graph. Then it puts back the removed edges one by one. Every time a new edge is put back the algorithm updates the colouring of the graph so that the colouring remains random. The performance of the algorithm depends heavily on certain spatial correlation decay properties of the Gibbs distribution

Sampling Random Colorings of Sparse Random Graphs

We study the mixing properties of the single-site Markov chain known as the Glauber dynamics for sampling $k$-colorings of a sparse random graph $G(n,d/n)$ for constant $d$. The best known rapid mixing results for general graphs are in terms of the maximum degree $\Delta$ of the input graph $G$ and hold when $k>11\Delta/6$ for all $G$. Improved results hold when $k>\alpha\Delta$ for graphs with girth $\geq 5$ and $\Delta$ sufficiently large where $\alpha\approx 1.7632\ldots$ is the root of $\alpha=\exp(1/\alpha)$; further improvements on the constant $\alpha$ hold with stronger girth and maximum degree assumptions. For sparse random graphs the maximum degree is a function of $n$ and the goal is to obtain results in terms of the expected degree $d$. The following rapid mixing results for $G(n,d/n)$ hold with high probability over the choice of the random graph for sufficiently large constant~$d$. Mossel and Sly (2009) proved rapid mixing for constant $k$, and Efthymiou (2014) improved this to $k$ linear in~$d$. The condition was improved to $k>3d$ by Yin and Zhang (2016) using non-MCMC methods. Here we prove rapid mixing when $k>\alpha d$ where $\alpha\approx 1.7632\ldots$ is the same constant as above. Moreover we obtain $O(n^{3})$ mixing time of the Glauber dynamics, while in previous rapid mixing results the exponent was an increasing function in $d$. As in previous results for random graphs our proof analyzes an appropriately defined block dynamics to "hide" high-degree vertices. One new aspect in our improved approach is utilizing so-called local uniformity properties for the analysis of block dynamics. To analyze the "burn-in" phase we prove a concentration inequality for the number of disagreements propagating in large blocks