arXiv subjects

Charilaos Efthymiou

Publications and source records attributed to Charilaos Efthymiou.

24 records · Page 2Linked to original sources

arXiv2013

Broadcasting colourings on trees. A combinatorial view

The broadcasting models on a d-ary tree T arise in many contexts such as biology, information theory, statistical physics and computer science. We consider the k-colouring model, i.e. the root of T is assigned an arbitrary colour and, conditional on this assignment, we take a random colouring of T. A basic question here is whether the information of the assignment at the root affects the distribution of the colourings at the leaves. This is the so-called reconstruction/non-reconstruction problem. It is well known that d/ln d is a threshold function for this problem, i.e. * if k \geq (1+\eps)d/ln d, then the colouring of the root has a vanishing effect on the distribution of the colourings at the leaves, as the height of the tree grows * if $k\leq (1-\eps)d/ln d, then the colouring of the root biases the distribution of the colouring of the leaves regardless of the height of the tree. There is no apparent combinatorial reason why such a result should be true. When k\geq (1+\eps)d/ ln d, the threshold implies the following: We can couple two broadcasting processes that assign the root different colours such that the probability of having disagreement at the leaves reduces with their distance from the root. It is natural to perceive such coupling as a mapping from the colouring of the first broadcasting process to the colouring of the second one. Here, we study how can we have such a mapping "combinatorially". Devising a mapping where the disagreements vanish as we move away from the root turns out to be a non-trivial task to accomplish for any k \leq d. In this work we obtain a coupling which has the aforementioned property for any k>3d/ln d, i.e. much smaller than d. Interestingly enough, the decisions that we make in the coupling are local. We relate our result to sampling k-colourings of sparse random graphs, with expected degree d and k<d.

cs.DM↗

arXiv2013

On the chromatic number of random regular graphs

Let G(n,d) be the random d-regular graph on n vertices. For any integer k exceeding a certain constant k_0 we identify a number d_{k-col} such that G(n,d) is k-colorable w.h.p. if d d_{k-col}.

math.CO↗

arXiv2013

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 $(1\pm n^{-Ω(1)})$approximation of the logarithm of the number of $k$-colourings of $G(n,d/n)$ for $k\geq (2+ε) d$ 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 $Λ$ of the graph by deleting edges between vertices of $Λ$. Then we show that the effect of this change on the marginals of Gibbs distribution, diminishes as we move away from $Λ$. Our approach is novel and suggests a new context for the study of deterministic counting algorithms.

cs.DM↗

arXiv2013

On independent sets in random graphs

The independence number of a sparse random graph G(n,m) of average degree d=2m/n is well-known to be α(G(n,m))~2n ln(d)/d with high probability. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size (1+o(1)) n ln(d)/d, i.e. half the maximum size. Yet in spite of 30 years of extensive research no efficient algorithm has emerged to produce an independent set with (1+c)n ln(d)/d, for any fixed c>0. In this paper we prove that the combinatorial structure of the independent set problem in random graphs undergoes a phase transition as the size k of the independent sets passes the point k nln(d)/d. Roughly speaking, we prove that independent sets of size k>(1+c)n ln(d)/d form an intricately ragged landscape, in which local search algorithms are bound to get stuck. We illustrate this phenomenon by providing an exponential lower bound for the Metropolis process, a Markov chain for sampling independents sets.

cs.DM↗

arXiv2011

A simple algorithm for random colouring G(n, d/n) using (2+ε)d colours

Approximate random k-colouring of a graph G=(V,E) is a very 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, i.e. the uniform distribution over all the k-colourings of G. Here, we deal with the problem when the underlying graph is an instance of Erdos-Renyi random graph G(n,p), where p=d/n and d is fixed. We propose a novel efficient algorithm for approximate random k-colouring with the following properties: given an instance of G(n,d/n) and for any k>(2+ε)d, it returns a k-colouring distributed within total variation distance n^{-Omega(1)} from the Gibbs distribution, with probability 1-n^{-Omega(1)}. What we propose is neither a MCMC algorithm nor some algorithm inspired by the message passing heuristics that were introduced by statistical physicist. Our algorithm is of combinatorial nature. It is based on a rather simple recursion which reduces the random k-colouring of G(n,d/n) to random k-colouring simpler subgraphs first. The lower bound on the number of colours for our algorithm to run in polynomial time is dramatically smaller than the corresponding bounds we have for any previous algorithm.

cs.DM↗

arXiv2008

Random sampling of colourings of sparse random graphs with a constant number of colours

In this work we present a simple and efficient algorithm which, with high probability, provides an almost uniform sample from the set of proper k-colourings on an instance of a sparse random graph G(n,d/n), where k=k(d) is a sufficiently large constant. Our algorithm is not based on the Markov Chain Monte Carlo method (M.C.M.C.). Instead, we provide a novel proof of correctness of our Algorithm that is based on interesting "spatial mixing" properties of colourings of G(n,d/n). Our result improves upon previous results (based on M.C.M.C.) that required a number of colours growing unboundedly with n.

cs.DM↗