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Algebraic Methods in Computational Complexity

Bläser, Markus and Kabanets, Valentine and Torán, Jacobo and Umans, Christopher (2018) Algebraic Methods in Computational Complexity. Dagstuhl Reports, 8 (9). pp. 133-153. ISSN 2192-5283. https://resolver.caltech.edu/CaltechAUTHORS:20191115-154547839

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Abstract

Computational Complexity is concerned with the resources that are required for algorithms to detect properties of combinatorial objects and structures. It has often proven true that the best way to argue about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The Razborov-Smolensky polynomial-approximation method for proving constant-depth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some of the most prominent examples. In some of the most exciting recent progress in Computational Complexity the algebraic theme still plays a central role. There have been significant recent advances in algebraic circuit lower bounds, and the so-called chasm at depth 4 suggests that the restricted models now being considered are not so far from ones that would lead to a general result. There have been similar successes concerning the related problems of polynomial identity testing and circuit reconstruction in the algebraic model (and these are tied to central questions regarding the power of randomness in computation). Also the areas of derandomization and coding theory have experimented important advances. The seminar aimed to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics and the goal of the seminar was to play an important role in educating a diverse community about the latest new techniques.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.4230/DagRep.8.9.133DOIArticle
https://www.dagstuhl.de/en/program/calendar/semhp/?semnr=18391OrganizationArticle
Additional Information:Creative Commons BY 3.0 Unported license. Report from Dagstuhl Seminar 18391.
Subject Keywords:computational complexity, algebra, (de-) randomization, circuits, coding, lower bounds
Issue or Number:9
Record Number:CaltechAUTHORS:20191115-154547839
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20191115-154547839
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:99875
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:16 Nov 2019 00:07
Last Modified:26 Nov 2019 21:42

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