Coupling with the stationary distribution and improved sampling for colorings and independent sets

We present an improved coupling technique for analyzing the mixing time of Markov chains. Using our technique, we simplify and extend previous results for sampling colorings and independent sets. Our approach uses properties of the stationary distribution to avoid worst-case configurations which arise in the traditional approach. As an application, we show that for $k/\Delta >1.764$, the Glauber dynamics on $k$-colorings of a graph on $n$ vertices with maximum degree $\Delta$ converges in $O(n\log n)$ steps, assuming $\Delta =\Omega(\log n)$ and that the graph is triangle-free. Previously, girth $\ge 5$ was needed. As a second application, we give a polynomial-time algorithm for sampling weighted independent sets from the Gibbs distribution of the hard-core lattice gas model at fugacity $\lambda <(1-\epsilon)e/\Delta$, on a regular graph $G$ on $n$ vertices of degree $\Delta =\Omega(\log n)$ and girth $\ge 6$. The best known algorithm for general graphs currently assumes $\lambda <2/(\Delta -2)$.Comment: Published at http://dx.doi.org/10.1214/105051606000000330 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

How to Wake up Your Neighbors: Safe and Nearly Optimal Generic Energy Conservation in Radio Networks

Recent work [Chang et al., 2018; Chang et al., 2020; Varsha Dani et al., 2021] has shown that it is sometimes feasible to significantly reduce the energy usage of some radio-network algorithms by adaptively powering down the radio receiver when it is not needed. Although past work has focused on modifying specific network algorithms in this way, we now ask the question of whether this problem can be solved in a generic way, treating the algorithm as a kind of black box. We are able to answer this question in the affirmative, presenting a new general way to modify arbitrary radio-network algorithms in an attempt to save energy. At the expense of a small increase in the time complexity, we can provably reduce the energy usage to an extent that is provably nearly optimal within a certain class of general-purpose algorithms. As an application, we show that our algorithm reduces the energy cost of breadth-first search in radio networks from the previous best bound of 2^O(?{log n}) to polylog(n), where n is the number of nodes in the network A key ingredient in our algorithm is hierarchical clustering based on additive Voronoi decomposition done at multiple scales. Similar clustering algorithms have been used in other recent work on energy-aware computation in radio networks, but we believe the specific approach presented here may be of independent interest