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Algebra in Computational Complexity

Agrawal, Manindra and Kabanets, Valentine and Thierauf, Thomas and Umans, Christopher (2014) Algebra in Computational Complexity. Dagstuhl Reports, 4 (9). pp. 85-105. ISSN 2192-5283. https://resolver.caltech.edu/CaltechAUTHORS:20191126-133603851

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Abstract

At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection 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. The algebraic theme continues in some of the most exciting recent progress in computational complexity. 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. Representation theory has emerged as an important tool in three separate lines of work: the "Geometric Complexity Theory" approach to P vs. NP and circuit lower bounds, the effort to resolve the complexity of matrix multiplication, and a framework for constructing locally testable codes. Coding theory has seen several algebraic innovations in recent years, including multiplicity codes, and new lower bounds. This seminar brought together researchers who are using a diverse array of algebraic methods in a variety of settings. It plays an important role in educating a diverse community about the latest new techniques, spurring further progress.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.4230/DagRep.4.9.85DOIArticle
http://www.dagstuhl.de/14391OrganizationArticle
Additional Information:Creative Commons BY 3.0 Unported license. Dagstuhl Seminar 14391.
Subject Keywords:Computational Complexity, lower bounds, approximazation, pseudorandomness, derandomization, circuits
Issue or Number:9
Record Number:CaltechAUTHORS:20191126-133603851
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20191126-133603851
Official Citation:1998 ACM Subject Classification: F.1.3 Complexity Measures and Classes, F.2 Analysis of Algorithms and Problem Complexity
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100070
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:26 Nov 2019 22:48
Last Modified:26 Nov 2019 22:48

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