Charilaos Efthymiou

Switching Colouring of G(n,d/n) for Sampling up to Gibbs Uniqueness Threshold

Approximate random k-colouring of a graph G = (V,E), efficiently, is a very well studied problem in computer science and statistical physics. It amounts to constructing, in polynomial time, a k-colouring of G which is distributed close to Gibbs distribution. Here, we deal with the problem when the underlying graph is an instance of Erdős-Renyi random graph G(n,d/n), where d is fixed.

Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model

We study the hard-core (gas) model defined on independent sets of an input graph where the independent sets are weighted by a parameter (aka fugacity) λ > 0. For constant Δ, previous work of Weitz (2006) established an FPTAS for the partition function for graphs of maximum degree Δ when λ <; λ_c(Δ). Sly (2010) showed that there is no FPRAS, unless NP=RP, when λ > λ_c(Δ). The threshold λ_c(Δ) is the critical point for the statistical physics phase transition for uniqueness/non-uniqueness on the infinite Δ-regular tree. The running time of Weitzs algorithm is exponential in log Δ. Here we present an FPRAS for the partition function whose running time is O* (n²). We analyze the simple single-site Markov chain known as the Glauber dynamics for sampling from the associated Gibbs distribution. We prove there exists a constant Δ₀ such that for all graphs with maximum degree Δ > Δ₀ and girth > 7 (i.e., no cycles of length ≤ 6), the mixing time of the Glauber dynamics is O(nlog n) when λ <; λ_c(Δ). Our work complements that of Weitz which applies for small constant Δ whereas our work applies for all Δ at least a sufficiently large constant Δ₀ (this includes Δ depending on n = IVI). Our proof utilizes loopy BP (belief propagation) which is a widely-used algorithm for inference in graphical models. A novel aspect of our work is using the principal eigenvector for the BP operator to design a distance function which contracts in expectation for pairs of states that behave like the BP fixed point. We also prove that the Glauber dynamics behaves locally like loopy BP. As a byproduct we obtain that the Glauber dynamics, after a short burn-in period, converges close to the BP fixed point, and this implies that the fixed point of loopy BP is a close approximation to the Gibbs distribution. Using these connections we establish that loopy BP quickly converges to the Gibbs distribution when the girth ≥ 6 and λ <; λ_c(Δ).

On independent sets in random graphs

The independence number of a sparse random graph <i>G(n, m)</i> of average degree <i>d</i> = 2<i>m/n</i> is well-known to be α<i>(G(n, m))</i> ~ 2<i>n</i> ln<i>(d)/d</i> with high probability. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size (1 + <i>o</i>(1)) · <i>n</i> ln<i>(d)/d</i>, 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 + ε)<i>n</i> ln<i>(d)/d</i>, for any fixed ε > 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 <i>k</i> of the independent sets passes the point <i>k ~ n</i> ln<i>(d)/d.</i> Roughly speaking, we prove that independent sets of size <i>k</i> > (1 + ε)<i>n</i> ln<i>(d)/d</i> 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.

Local Convergence of Random Graph Colorings

Let

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

G=G(n,m)

Sharp thresholds for Hamiltonicity in random intersection graphs

be a random graph whose average degree

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

d=2m/n

Planting Colourings Silently

is below the

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

k

Sampling random colorings of sparse random graphs

-colorability threshold. If we sample a