Document

**Published in:** Dagstuhl Reports, Volume 8, Issue 9 (2019)

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.

Markus Bläser, Valentine Kabanets, Jacobo Torán, and Christopher Umans. Algebraic Methods in Computational Complexity (Dagstuhl Seminar 18391). In Dagstuhl Reports, Volume 8, Issue 9, pp. 133-153, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Article{blaser_et_al:DagRep.8.9.133, author = {Bl\"{a}ser, Markus and Kabanets, Valentine and Tor\'{a}n, Jacobo and Umans, Christopher}, title = {{Algebraic Methods in Computational Complexity (Dagstuhl Seminar 18391)}}, pages = {133--153}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2019}, volume = {8}, number = {9}, editor = {Bl\"{a}ser, Markus and Kabanets, Valentine and Tor\'{a}n, Jacobo and Umans, Christopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.8.9.133}, URN = {urn:nbn:de:0030-drops-103438}, doi = {10.4230/DagRep.8.9.133}, annote = {Keywords: computational complexity, algebra, (de-) randomization, circuits, coding, lower bounds} }

Document

**Published in:** Dagstuhl Reports, Volume 6, Issue 10 (2017)

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.
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).
Another surprising connection is that the algebraic techniques invented to show lower bounds now prove useful to develop efficient algorithms. For example,
Williams showed how to use the polynomial method to obtain faster all-pair-shortest-path algorithms. This emphases once again the central role of algebra in computer science.
The seminar aims 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 this seminar can play an important role in educating a diverse community about the latest new techniques, spurring further progress.

Valentine Kabanets, Thomas Thierauf, Jacobo Tóran, and Christopher Umans. Algebraic Methods in Computational Complexity (Dagstuhl Seminar 16411). In Dagstuhl Reports, Volume 6, Issue 10, pp. 13-32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@Article{kabanets_et_al:DagRep.6.10.13, author = {Kabanets, Valentine and Thierauf, Thomas and T\'{o}ran, Jacobo and Umans, Christopher}, title = {{Algebraic Methods in Computational Complexity (Dagstuhl Seminar 16411)}}, pages = {13--32}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2017}, volume = {6}, number = {10}, editor = {Kabanets, Valentine and Thierauf, Thomas and T\'{o}ran, Jacobo and Umans, Christopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.6.10.13}, URN = {urn:nbn:de:0030-drops-69504}, doi = {10.4230/DagRep.6.10.13}, annote = {Keywords: Computational Complexity, lower bounds, approximation, pseudo-randomness, derandomization, circuits} }

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**Published in:** LIPIcs, Volume 61, 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)

A line of work initiated by Terhal and DiVincenzo [Terhal/DiVincenzo, arXiv, 2002] and Bremner, Jozsa, and Shepherd [Bremner/Jozsa/Sheperd, Proc. Royal Soc. A, 2010], shows that restricted classes of quantum computation can efficiently sample from probability distributions that cannot be exactly sampled efficiently on a classical computer, unless the PH collapses. Aaronson and Arkhipov [Aaronson/Arkhipov, J. Theory of Comp., 2013] take this further by considering a distribution that can be sampled efficiently by linear optical quantum computation, that under two feasible conjectures, cannot even be approximately sampled within bounded total variation distance, unless the PH collapses.
In this work we use Quantum Fourier Sampling to construct a class of distributions that can be sampled exactly by a quantum computer. We then argue that these distributions cannot be approximately sampled classically, unless the PH collapses, under variants of the Aaronson-Arkhipov conjectures.
In particular, we show a general class of quantumly sampleable distributions each of which is based on an "Efficiently Specifiable" polynomial, for which a classical approximate sampler implies an average-case approximation. This class of polynomials contains the Permanent but also includes, for example, the Hamiltonian Cycle polynomial, as well as many other familiar #P-hard polynomials.
Since our distribution likely requires the full power of universal quantum computation, while the Aaronson-Arkhipov distribution uses only linear optical quantum computation with noninteracting bosons, why is our result interesting? We can think of at least three reasons:
1. Since the conjectures required in [Aaronson/Arkhipov, J. Theory of Comp., 2013] have not yet been proven, it seems worthwhile to weaken them as much as possible. We do this in two ways, by weakening both conjectures to apply to any "Efficiently Specifiable" polynomial, and by weakening the so-called Anti-Concentration conjecture so that it need only hold for one distribution in a broad class of distributions.
2. Our construction can be understood without any knowledge of linear optics. While this may be a disadvantage for experimentalists, in our opinion it results in a very clean and simple exposition that may be more immediately accessible to computer scientists.
3. It is extremely common for quantum computations to employ “Quantum Fourier Sampling” in the following way: first apply a classically efficient function to a uniform superposition of inputs, then apply a Quantum Fourier Transform followed by a measurement. Our distributions are obtained in exactly this way, where the classically efficient function is related to a (presumed) hard polynomial. Establishing rigorously a robust sense in which the central primitive of Quantum Fourier Sampling is classically hard seems a worthwhile goal in itself.

Bill Fefferman and Christopher Umans. On the Power of Quantum Fourier Sampling. In 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 61, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fefferman_et_al:LIPIcs.TQC.2016.1, author = {Fefferman, Bill and Umans, Christopher}, title = {{On the Power of Quantum Fourier Sampling}}, booktitle = {11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)}, pages = {1:1--1:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-019-4}, ISSN = {1868-8969}, year = {2016}, volume = {61}, editor = {Broadbent, Anne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2016.1}, URN = {urn:nbn:de:0030-drops-66829}, doi = {10.4230/LIPIcs.TQC.2016.1}, annote = {Keywords: Quantum Complexity Theory, Sampling Complexity} }

Document

**Published in:** Dagstuhl Reports, Volume 4, Issue 9 (2015)

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.

Manindra Agrawal, Valentine Kabanets, Thomas Thierauf, and Christopher Umans. Algebra in Computational Complexity (Dagstuhl Seminar 14391). In Dagstuhl Reports, Volume 4, Issue 9, pp. 85-105, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@Article{agrawal_et_al:DagRep.4.9.85, author = {Agrawal, Manindra and Kabanets, Valentine and Thierauf, Thomas and Umans, Christopher}, title = {{Algebra in Computational Complexity (Dagstuhl Seminar 14391)}}, pages = {85--105}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2015}, volume = {4}, number = {9}, editor = {Agrawal, Manindra and Kabanets, Valentine and Thierauf, Thomas and Umans, Christopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.4.9.85}, URN = {urn:nbn:de:0030-drops-48851}, doi = {10.4230/DagRep.4.9.85}, annote = {Keywords: Computational Complexity, lower bounds, approximazation, pseudo-randomness, derandomization, circuits} }

Document

**Published in:** Dagstuhl Reports, Volume 2, Issue 10 (2013)

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 (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 PCP characterization of NP and the
Agrawal-Kayal-Saxena polynomial-time primality test are two prominent examples.
Recently, there have been some works going in the opposite direction, giving alternative combinatorial proofs for results that were originally proved
algebraically. These alternative proofs can yield important improvements because they are closer to the underlying problems and avoid the losses in passing to the algebraic setting. A prominent example is Dinur's proof of the PCP Theorem via gap amplification which yielded short PCPs with only a polylogarithmic length blowup (which had been the focus of significant research effort up to that point). We see here (and in a number of recent works) an exciting interplay between algebraic and combinatorial techniques.
This seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic and combinatorial methods in a variety of settings.

Manindra Agrawal, Thomas Thierauf, and Christopher Umans. Algebraic and Combinatorial Methods in Computational Complexity (Dagstuhl Seminar 12421). In Dagstuhl Reports, Volume 2, Issue 10, pp. 60-78, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@Article{agrawal_et_al:DagRep.2.10.60, author = {Agrawal, Manindra and Thierauf, Thomas and Umans, Christopher}, title = {{Algebraic and Combinatorial Methods in Computational Complexity (Dagstuhl Seminar 12421)}}, pages = {60--78}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2013}, volume = {2}, number = {10}, editor = {Agrawal, Manindra and Thierauf, Thomas and Umans, Christopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.2.10.60}, URN = {urn:nbn:de:0030-drops-39034}, doi = {10.4230/DagRep.2.10.60}, annote = {Keywords: Computational Complexity, lower bounds, approximazation, pseudo-randomness, derandomization, circuits} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 9421, Algebraic Methods in Computational Complexity (2010)

From 11.10. to 16.10.2009, the Dagstuhl Seminar 09421 ``Algebraic Methods in Computational Complexity '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Manindra Agrawal, Lance Fortnow, Thomas Thierauf, and Christopher Umans. 09421 Abstracts Collection – Algebraic Methods in Computational Complexity. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 9421, pp. 1-22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{agrawal_et_al:DagSemProc.09421.1, author = {Agrawal, Manindra and Fortnow, Lance and Thierauf, Thomas and Umans, Christopher}, title = {{09421 Abstracts Collection – Algebraic Methods in Computational Complexity}}, booktitle = {Algebraic Methods in Computational Complexity}, pages = {1--22}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {9421}, editor = {Manindra Agrawal and Lance Fortnow and Thomas Thierauf and Christopher Umans}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09421.1}, URN = {urn:nbn:de:0030-drops-24181}, doi = {10.4230/DagSemProc.09421.1}, annote = {Keywords: Computational Complexity, Algebra} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 9421, Algebraic Methods in Computational Complexity (2010)

The seminar brought together more than 50 researchers covering
a wide spectrum of complexity theory. The focus on algebraic
methods showed once again the great importance of algebraic
techniques for theoretical computer science. We had almost 30
talks, most of them about 40 minutes leaving ample room for
discussions. We also had a much appreciated open problem
session.
The talks ranged over a
broad assortment of subjects with the underlying theme of using
algebraic techniques. It was very fruitful and has hopefully
initiated new directions in research. Several participants
specifically mentioned that they appreciated the particular
focus on a common class of techniques (rather than end
results) as a unifying theme of the workshop. We look forward
to our next meeting!

Manindra Agrawal, Lance Fortnow, Thomas Thierauf, and Christopher Umans. 09421 Executive Summary – Algebraic Methods in Computational Complexity. In Algebraic Methods in Computational Complexity. Dagstuhl Seminar Proceedings, Volume 9421, pp. 1-4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{agrawal_et_al:DagSemProc.09421.2, author = {Agrawal, Manindra and Fortnow, Lance and Thierauf, Thomas and Umans, Christopher}, title = {{09421 Executive Summary – Algebraic Methods in Computational Complexity}}, booktitle = {Algebraic Methods in Computational Complexity}, pages = {1--4}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {9421}, editor = {Manindra Agrawal and Lance Fortnow and Thomas Thierauf and Christopher Umans}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09421.2}, URN = {urn:nbn:de:0030-drops-24100}, doi = {10.4230/DagSemProc.09421.2}, annote = {Keywords: Computational Complexity, Algebra} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 8381, Computational Complexity of Discrete Problems (2008)

We obtain randomized algorithms for factoring degree $n$
univariate polynomials over $F_q$ requiring $O(n^{1.5 +
o(1)} log^{1+o(1)} q+ n^{1 + o(1)}log^{2+o(1)} q)$ bit operations.
When $log q < n$, this is asymptotically faster than the best previous algorithms (von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998)); for
$log q ge n$, it matches the asymptotic running time of the best
known algorithms.
The improvements come from new algorithms for modular composition
of degree $n$ univariate polynomials, which is the asymptotic
bottleneck in fast algorithms for factoring polynomials over
finite fields. The best previous algorithms for modular
composition use $O(n^{(omega + 1)/2})$ field operations, where
$omega$ is the exponent of matrix multiplication (Brent & Kung
(1978)), with a slight improvement in the exponent achieved by
employing fast rectangular matrix multiplication (Huang & Pan
(1997)).
We show that modular composition and multipoint evaluation of
multivariate polynomials are essentially equivalent, in the sense
that an algorithm for one achieving exponent $alpha$ implies an
algorithm for the other with exponent $alpha + o(1)$, and vice
versa. We then give two new algorithms that solve the problem
optimally (up to lower order terms): an algebraic algorithm for
fields of characteristic at most $n^{o(1)}$, and a
nonalgebraic algorithm that works in arbitrary characteristic.
The latter algorithm works by lifting to characteristic 0,
applying a small number of rounds of {em multimodular reduction},
and finishing with a small number of multidimensional FFTs. The
final evaluations are reconstructed using the Chinese Remainder
Theorem. As a bonus, this algorithm produces a very efficient data
structure supporting polynomial evaluation queries, which is of
independent interest.
Our algorithms use techniques which are commonly employed in
practice, so they may be competitive for real problem sizes. This
contrasts with all previous subquadratic algorithsm for these
problems, which rely on fast matrix multiplication.
This is joint work with Kiran Kedlaya.

Kiran Kedlaya and Christopher Umans. Fast polynomial factorization and modular composition. In Computational Complexity of Discrete Problems. Dagstuhl Seminar Proceedings, Volume 8381, pp. 1-33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{kedlaya_et_al:DagSemProc.08381.5, author = {Kedlaya, Kiran and Umans, Christopher}, title = {{Fast polynomial factorization and modular composition}}, booktitle = {Computational Complexity of Discrete Problems}, pages = {1--33}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8381}, editor = {Peter Bro Miltersen and R\"{u}diger Reischuk and Georg Schnitger and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08381.5}, URN = {urn:nbn:de:0030-drops-17771}, doi = {10.4230/DagSemProc.08381.5}, annote = {Keywords: Modular composition; polynomial factorization; multipoint evaluation; Chinese Remaindering} }