AF: Medium: Collaborative Research: Information Compression in Algorithm Design and Statistical Physics

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The existence of connections between probabilistic algorithms, statistical physics and information theory has been known for decades and has yielded a number of unexpected breakthroughs. Recent discoveries of the PIs and other researchers give clear indications that these connections go much deeper than previously thought. A key new idea is the realization that stochastic local search algorithms can be judged by their capacity to compress the randomness they consume, with convergence following as a consequence of compressibility. Further exploration of this idea is expected to have significant impact, both conceptual and technical, in multiple scientific fields. This includes algorithm design by information theoretic methods, the study of phase transitions in statistical mechanical systems based on information bottleneck arguments, and non-constructive proofs of existence of combinatorial objects. The project will offer a wide range of research opportunities at various levels of sophistication for graduate and undergraduate students in three state universities.Information compression arguments have recently found striking applications in computer science and combinatorics. A glowing example is Moser's proof of the algorithmic Lovasz Local Lemma, which suggested an entirely new way of reasoning about randomized algorithms. Inspired by the work of Moser, one of the PIs with a collaborator has very recently created a general framework for analyzing stochastic local search algorithms using information compression. The framework is purely algorithmic, completely bypassing the Probabilistic Method. Besides helping to analyze the running times of existing algorithms it can also be used as a powerful new tool for designing novel, non-obvious randomized algorithms. The proposed research further develops this framework with the aim of unearthing completely new applications in computer science and combinatorics, while establishing mathematically rigorous connections to statistical physics. Concrete examples of such applications to be investigated include new tools for bounding the mixing time of Markov chains and algebraic connections between randomized algorithms and the classical theory of phase transitions in statistical physics.