Eric Vigoda

Improved bounds for sampling colorings

We consider the problem of sampling uniformly at random from the set of proper k-colorings of a graph with maximum degree Δ. Our main result is the design of a simple Markov chain that converges in O(nk log n) time to the desired distribution when k>116Δ.

Heterogeneous Genomic Molecular Clocks in Primates

Using data from primates, we show that molecular clocks in sites that have been part of a CpG dinucleotide in recent past (CpG sites) and non-CpG sites are of markedly different nature, reflecting differences in their molecular origins. Notably, single nucleotide substitutions at non-CpG sites show clear generation-time dependency, indicating that most of these substitutions occur by errors during DNA replication. On the other hand, substitutions at CpG sites occur relatively constantly over time, as expected from their primary origin due to methylation. Therefore, molecular clocks are heterogeneous even within a genome. Furthermore, we propose that varying frequencies of CpG dinucleotides in different genomic regions may have contributed significantly to conflicting earlier results on rate constancy of mammalian molecular clock. Our conclusion that different regions of genomes follow different molecular clocks should be considered when inferring divergence times using molecular data and in phylogenetic analysis.

Torpid mixing of some Monte Carlo Markov chain algorithms in statistical physics

Studies two widely used algorithms, Glauber dynamics and the Swendsen-Wang (1987) algorithm, on rectangular subsets of the hypercubic lattice Z/sup d/. We prove that, under certain circumstances, the mixing time in a box of side length L with periodic boundary conditions can be exponential in L/sup d-1/. In other words, under these circumstances, the mixing in these widely used algorithms is not rapid; instead it is torpid. The models we study are the independent set model and the q-state Potts model. For both models, we prove that Glauber dynamics is torpid in the region with phase coexistence. For the Potts model, we prove that the Swendsen-Wang mixing is torpid at the phase transition point.

A non-Markovian coupling for randomly sampling colorings

We study a simple Markov chain, known as the Glauber dynamics, for randomly sampling (proper) k-colorings of an input graph G on n vertices with maximum degree /spl Delta/ and girth g. We prove the Glauber dynamics is close to the uniform distribution after O(n log n) steps whenever k > (1 + /spl epsiv/)/spl Delta/, for all /spl epsiv/ > 0, assuming g /spl ges/ 9 and /spl Delta/ = /spl Omega/(log n). The best previously known bounds were k > 11/spl Delta//6 for general graphs, and k > 1.489/spl Delta/ for graphs satisfying girth and maximum degree requirements. Our proof relies on the construction and analysis of a non-Markovian coupling. This appears to be the first application of a non-Markovian coupling to substantially improve upon known results.

Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models

Recent inapproximability results of Sly ( 2010 ), together with an approximation algorithm presented by Weitz ( 2006 ), establish a beautiful picture of the computational complexity of approximating the partition function of the hard-core model. Let λ c (

Adaptive simulated annealing: A near-optimal connection between sampling and counting

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Limitations of Markov chain Monte Carlo algorithms for Bayesian inference of phylogeny

) denote the critical activity for the hard-model on the infinite Δ-regular tree. Weitz presented an FPTAS for the partition function when λ c (

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

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Negative examples for sequential importance sampling of binary contingency tables

) for graphs with constant maximum degree Δ. In contrast, Sly showed that for all Δ ⩾ 3, there exists e Δ > 0 such that (unless RP = NP) there is no FPRAS for approximating the partition function on graphs of maximum degree Δ for activities λ satisfying λ c (

An FPTAS for #Knapsack and Related Counting Problems

\mathbb{T}_{\Delta}