Randomness has emerged as a core concept and tool in computation. From modeling phenomena to efficient algorithms to proof techniques, the applications of randomness are ubiquitous and powerful. Notable examples include: construction of important combinatorial objects such as expanders, rigorously establishing phase transitions in physical models, finding polynomial-time algorithms for fundamental sampling problems and approximating #P-hard counting problems, designing probabilistically checkable proofs (PCP's) and establishing the hardness of approximation, and discovering simpler and often faster algorithms for a variety of computational problems. In the course of these tremendous developments, several general-purpose techniques have emerged, and random sampling has become a fundamental, universal tool across sciences, engineering and computation.
This project brings together leading researchers in randomized algorithms to solve hard problems in random sampling, to identify techniques, and to develop new analytical tools. The applications come from a range of fields, including complexity, physics, biology, operations research and mathematics. The most general and widely-studied technique for sampling is simulating a Markov chain by taking a random walk on a suitable state space. The Markov Chain method and its application to sampling, counting and integration, broadly known as the Markov Chain Monte Carlo (MCMC) method, is a central theme of the project.
Intellectual Merit. The project focuses on applications of randomized algorithms and random sampling to rigorously address problems across several disciplines. Within computer science these topics include: massive data sets, where sampling is critical both for finding low-dimensional representations and clustering; routing networks, where sampling has many applications from monitoring and path allocation to optimization; machine learning; and property testing. Recent interactions between computer science and other scientic disciplines have led to many new rigorous applications of sampling, as well as new insights in how to design and analyze efficient algorithms with performance guarantees; for instance, phase transitions in the underlying physical models can cause local Markov chains to be inefficient. The project explores deeper connections between physics and random sampling, including conjectured correlations between reconstruction problems and thresholds for the efficiency of local algorithms. Many related problems arise in biology, such as phylogenetic tree reconstruction and analysis of complex biological networks. In nanotechology, models of self-assembly are simple Markov chains. In mathematics, the techniques used in the analysis of sampling algorithms in general and Markov chains in particular have drawn heavily on probability theory, both discrete and continuous.
Broader Impact. The college of computing at Georgia Tech is home to the new Algorithms and Randomness Center (ARC) with many faculty and students sharing this expertise. The project's activities include designing a summer school for graduate students in randomized algorithms and designing a course for training students from diverse backgrounds and hosting workshops focusing on both theoretical and applied aspects of randomized algorithms. Participation of women and under-represented groups in all of these activities will be encouraged, and the workshops will include tutorials to increase accessibility. These coordinated efforts in education and research will solidify the impact of ARC and make it a premier center for algorithms, randomness and complexity.
