Georgia Institute of Technology College of Sciences

At the intersection of computability and algebraic geometry, the following question arises: does an integral polynomial system of equations have any integral solutions? Famously, the combined work of Robinson, Davis, Putnam, and Matiyasevich answers this in the negative. Nonetheless, algorithms have played in increasing role in the development of algebraic geometry and its many applications. I address some research related to this general theme and some outstanding questions.

We study the cost function for hierarchical clusterings introduced by Dasgupta where hierarchies are treated as first-class objects rather than deriving their cost from projections into flat clusters. It was also shown that a top-down algorithm returns a hierarchical clustering of cost at most O (α_n log n) times the cost of the optimal hierarchical clustering, where α_n is the approximation ratio of the Sparsest Cut subroutine used. Thus using the best known approximation algorithm for Sparsest Cut due to Arora-Rao-Vazirani, the top down algorithm returns a hierarchical clustering of cost at most O(log^{3/2} n) times the cost of the optimal solution. We improve this by giving an O(log n)- approximation algorithm for this problem. Our main technical ingredients are a combinatorial characterization of ultrametrics induced by this cost function, deriving an Integer Linear Programming (ILP) formulation for this family of ultrametrics, and showing how to iteratively round an LP relaxation of this formulation by using the idea of sphere growing which has been extensively used in the context of graph partitioning. We also prove that our algorithm returns an O(log n)-approximate hierarchical clustering for a generalization of this cost function also studied in Dasgupta. This joint work with Sebastian Pokutta is to appear in NIPS 2016 (oral presentation).

Graded posets are partially ordered sets equipped with a unique rank function that respects the partial order and such that neighboring elements in the Hasse diagram have ranks that differ by one. We frequently find them throughout combinatorics, including the canonical partial order on Young diagrams and plane partitions, where their respective rank functions are the area and volume under the configuration. We ask when it is possible to efficiently sample elements with a fixed rank in a graded poset. We show that for certain classes of posets, a biased Markov chain that connects elements in the Hasse diagram allows us to approximately generate samples from any fixed rank in expected polynomial time. While varying a bias parameter to increase the likelihood of a sample of a desired size is common in statistical physics, one typically needs properties such as log-concavity in the number of elements of each size to generate desired samples with sufficiently high probability. Here we do not even require unimodality in order to guarantee that the algorithm succeeds in generating samples of the desired rank efficiently. This joint work with Prateek Bhakta, Ben Cousins, and Dana Randall will appear at SODA 2017.

I will present work on a new application of Markov chains to distributed computing. Motivated by programmable matter and the behavior of biological distributed systems such as ant colonies, the geometric amoebot model abstracts these processes as self-organizing particle systems where particles with limited computational power move on the triangular lattice. Previous algorithms developed in this setting have relied heavily on leader election, tree structures that are not robust to failures, and persistent memory. We developed a distributed algorithm for the compression problem, where all particles want to gather together as tightly as possible, that is based on a Markov chain and is simple, robust, and oblivious. Tools from Markov chain analysis enable rigorous proofs about its behavior, and we show compression will occur with high probability. This joint work with Joshua J. Daymude, Dana Randall, and Andrea Richa appeared at PODC 2016. I will also present some more recent extensions of this approach to other problems, which is joint work with Marta Andres Arroyo as well.