Georgia Institute of Technology College of Sciences

The Swendsen-Wang dynamics is a popular algorithm for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. The dynamics is a global Markov chain that is conjectured to converge quickly to equilibrium even at low temperatures, where the correlations in the system are strong and local chains converge slowly. In this talk, we present new results concerning the speed of convergence of the Swendsen-Wang dynamics under spatial mixing (i.e., decay of correlations) conditions. In particular, we provide tight results for three distinct geometries: the integer d-dimensional integer lattice graph Z^d, regular trees, and random d-regular graphs. Our approaches crucially exploit the underlying geometry in different ways in each case.

We present a new rounding framework and improve the approximation bounds for min sum vertex cover and generalized min sum set cover, also known as multiple intents re-ranking problem. These classical combinatorial optimization problems find applications in scheduling, speeding up semidefinite-program solvers, and query-results diversification, among others.

Our algorithm is based on transforming the LP solution by a suitable kernel and applying a randomized rounding. It also gives a linear-programming based 4-approximation algorithm for min sum set cover, i.e., best possible due to Feige, Lovász, and Tetali. As part of the analysis of our randomized algorithm we derive an inequality on the lower tail of a sum of independent Bernoulli random variables, which may be of independent interest.

Joint work with Nikhil Bansal, Jatin Batra, and Prasad Tetali. [arXiv:2007.09172]

Hadwiger's conjecture from 1943 states that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the early 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable.  In this talk, we show that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable for every $\beta > 1/4$, making the first improvement on the order of magnitude of the Kostochka-Thomason bound. 

This is joint work with  Sergey Norin and Luke Postle.