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Documents authored by Bläser, Markus

Document
DOI: 10.4230/DagRep.12.9.41
Algebraic and Analytic Methods in Computational Complexity (Dagstuhl Seminar 22371)

Authors: Markus Bläser, Valentine Kabanets, Ronen Shaltiel, and Jacobo Torán

Published in: Dagstuhl Reports, Volume 12, Issue 9 (2023)

Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 2237 "Algebraic and Analytic Methods in Computational Complexity". 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. Beside algebraic methods, analytic methods have been used for quite some time in theoretical computer science. These methods can also be used to solve algebraic problems as show by many recent examples in the areas of derandomization, coding theory or circuit lower bounds. These new directions were in the focus of the Dagstuhl Seminar and reflect the developments in the field since the previous Dagstuhl Seminar 18391. This Dagstuhl Seminar brought together researchers who are using a diverse array of algebraic and analytic methods in a variety of settings. Researchers in these areas are relying on ever more sophisticated and specialized mathematics and this seminar played a role in educating a diverse community about the latest new techniques, spurring further progress.
Cite as

Markus Bläser, Valentine Kabanets, Ronen Shaltiel, and Jacobo Torán. Algebraic and Analytic Methods in Computational Complexity (Dagstuhl Seminar 22371). In Dagstuhl Reports, Volume 12, Issue 9, pp. 41-59, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{blaser_et_al:DagRep.12.9.41,
  author =	{Bl\"{a}ser, Markus and Kabanets, Valentine and Shaltiel, Ronen and Tor\'{a}n, Jacobo},
  title =	{{Algebraic and Analytic Methods in Computational Complexity (Dagstuhl Seminar 22371)}},
  pages =	{41--59},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2023},
  volume =	{12},
  number =	{9},
  editor =	{Bl\"{a}ser, Markus and Kabanets, Valentine and Shaltiel, Ronen and Tor\'{a}n, Jacobo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagRep.12.9.41},
  URN =		{urn:nbn:de:0030-drops-178092},
  doi =		{10.4230/DagRep.12.9.41},
  annote =	{Keywords: (de-)randomization, algebra, circuits, coding, computational complexity}
}
Document
DOI: 10.4230/LIPIcs.STACS.2023.12
On the Multilinear Complexity of Associative Algebras

Authors: Markus Bläser, Hendrik Mayer, and Devansh Shringi

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract
Christandl and Zuiddam [Matthias Christandl and Jeroen Zuiddam, 2019] study the multilinear complexity of d-fold matrix multiplication in the context of quantum communication complexity. Bshouty [Nader H. Bshouty, 2013] investigates the multilinear complexity of d-fold multiplication in commutative algebras to understand the size of so-called testers. The study of bilinear complexity is a classical topic in algebraic complexity theory, starting with the work by Strassen. However, there has been no systematic study of the multilinear complexity of multilinear maps. In the present work, we systematically investigate the multilinear complexity of d-fold multiplication in arbitrary associative algebras. We prove a multilinear generalization of the famous Alder-Strassen theorem, which is a lower bound for the bilinear complexity of the (2-fold) multiplication in an associative algebra. We show that the multilinear complexity of the d-fold multiplication has a lower bound of d ⋅ dim A - (d-1)t, where t is the number of maximal twosided ideals in A. This is optimal in the sense that there are algebras for which this lower bound is tight. Furthermore, we prove the following dichotomy that the quotient algebra A/rad A determines the complexity of the d-fold multiplication in A: When the semisimple algebra A/rad A is commutative, then the multilinear complexity of the d-fold multiplication in A is polynomial in d. On the other hand, when A/rad A is noncommutative, then the multilinear complexity of the d-fold multiplication in A is exponential in d.
Cite as

Markus Bläser, Hendrik Mayer, and Devansh Shringi. On the Multilinear Complexity of Associative Algebras. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blaser_et_al:LIPIcs.STACS.2023.12,
  author =	{Bl\"{a}ser, Markus and Mayer, Hendrik and Shringi, Devansh},
  title =	{{On the Multilinear Complexity of Associative Algebras}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{12:1--12:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.12},
  URN =		{urn:nbn:de:0030-drops-176645},
  doi =		{10.4230/LIPIcs.STACS.2023.12},
  annote =	{Keywords: Multilinear computations, associative algebras, matrix multiplication, Alder-Strassen theorem}
}
Document
DOI: 10.4230/LIPIcs.CCC.2021.29
On the Complexity of Evaluating Highest Weight Vectors

Authors: Markus Bläser, Julian Dörfler, and Christian Ikenmeyer

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract
Geometric complexity theory (GCT) is an approach towards separating algebraic complexity classes through algebraic geometry and representation theory. Originally Mulmuley and Sohoni proposed (SIAM J Comput 2001, 2008) to use occurrence obstructions to prove Valiant’s determinant vs permanent conjecture, but recently Bürgisser, Ikenmeyer, and Panova (Journal of the AMS 2019) proved this impossible. However, fundamental theorems of algebraic geometry and representation theory grant that every lower bound in GCT can be proved by the use of so-called highest weight vectors (HWVs). In the setting of interest in GCT (namely in the setting of polynomials) we prove the NP-hardness of the evaluation of HWVs in general, and we give efficient algorithms if the treewidth of the corresponding Young-tableau is small, where the point of evaluation is concisely encoded as a noncommutative algebraic branching program! In particular, this gives a large new class of separating functions that can be efficiently evaluated at points with low (border) Waring rank. As a structural side result we prove that border Waring rank is bounded from above by the ABP width complexity.
Cite as

Markus Bläser, Julian Dörfler, and Christian Ikenmeyer. On the Complexity of Evaluating Highest Weight Vectors. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 29:1-29:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blaser_et_al:LIPIcs.CCC.2021.29,
  author =	{Bl\"{a}ser, Markus and D\"{o}rfler, Julian and Ikenmeyer, Christian},
  title =	{{On the Complexity of Evaluating Highest Weight Vectors}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{29:1--29:36},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.29},
  URN =		{urn:nbn:de:0030-drops-143036},
  doi =		{10.4230/LIPIcs.CCC.2021.29},
  annote =	{Keywords: Algebraic complexity theory, geometric complexity theory, algebraic branching program, Waring rank, border Waring rank, representation theory, highest weight vector, treewidth}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.STACS.2021
LIPIcs, Volume 187, STACS 2021, Complete Volume

Authors: Markus Bläser and Benjamin Monmege

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
LIPIcs, Volume 187, STACS 2021, Complete Volume
Cite as

38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 1-988, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@Proceedings{blaser_et_al:LIPIcs.STACS.2021,
  title =	{{LIPIcs, Volume 187, STACS 2021, Complete Volume}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{1--988},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021},
  URN =		{urn:nbn:de:0030-drops-136444},
  doi =		{10.4230/LIPIcs.STACS.2021},
  annote =	{Keywords: LIPIcs, Volume 187, STACS 2021, Complete Volume}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.STACS.2021.0
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Markus Bläser and Benjamin Monmege

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as

38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blaser_et_al:LIPIcs.STACS.2021.0,
  author =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{0:i--0:xvi},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.0},
  URN =		{urn:nbn:de:0030-drops-136459},
  doi =		{10.4230/LIPIcs.STACS.2021.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.17
Slice Rank of Block Tensors and Irreversibility of Structure Tensors of Algebras

Authors: Markus Bläser and Vladimir Lysikov

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
Determining the exponent of matrix multiplication ω is one of the central open problems in algebraic complexity theory. All approaches to design fast matrix multiplication algorithms follow the following general pattern: We start with one "efficient" tensor T of fixed size and then we use a way to get a large matrix multiplication out of a large tensor power of T. In the recent years, several so-called barrier results have been established. A barrier result shows a lower bound on the best upper bound for the exponent of matrix multiplication that can be obtained by a certain restriction starting with a certain tensor. We prove the following barrier over C: Starting with a tensor of minimal border rank satisfying a certain genericity condition, except for the diagonal tensor, it is impossible to prove ω = 2 using arbitrary restrictions. This is astonishing since the tensors of minimal border rank look like the most natural candidates for designing fast matrix multiplication algorithms. We prove this by showing that all of these tensors are irreversible, using a structural characterisation of these tensors. To obtain our result, we relate irreversibility to asymptotic slice rank and instability of tensors and prove that the instability of block tensors can often be decided by looking only on the sizes of nonzero blocks.
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Markus Bläser and Vladimir Lysikov. Slice Rank of Block Tensors and Irreversibility of Structure Tensors of Algebras. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{blaser_et_al:LIPIcs.MFCS.2020.17,
  author =	{Bl\"{a}ser, Markus and Lysikov, Vladimir},
  title =	{{Slice Rank of Block Tensors and Irreversibility of Structure Tensors of Algebras}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{17:1--17:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.17},
  URN =		{urn:nbn:de:0030-drops-126869},
  doi =		{10.4230/LIPIcs.MFCS.2020.17},
  annote =	{Keywords: Tensors, Slice rank, Barriers, Matrix multiplication, GIT stability}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.8
Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness

Authors: Markus Bläser and Anurag Pandey

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract
We give a randomized polynomial time algorithm for polynomial identity testing for the class of n-variate poynomials of degree bounded by d over a field 𝔽, in the blackbox setting. Our algorithm works for every field 𝔽 with | 𝔽 | ≥ d+1, and uses only d log n + log (1/ ε) + O(d log log n) random bits to achieve a success probability 1 - ε for some ε > 0. In the low degree regime that is d ≪ n, it hits the information theoretic lower bound and differs from it only in the lower order terms. Previous best known algorithms achieve the number of random bits (Guruswami-Xing, CCC'14 and Bshouty, ITCS'14) that are constant factor away from our bound. Like Bshouty, we use Sidon sets for our algorithm. However, we use a new construction of Sidon sets to achieve the improved bound. We also collect two simple constructions of hitting sets with information theoretically optimal size against the class of n-variate, degree d polynomials. Our contribution is that we give new, very simple proofs for both the constructions.
Cite as

Markus Bläser and Anurag Pandey. Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{blaser_et_al:LIPIcs.APPROX/RANDOM.2020.8,
  author =	{Bl\"{a}ser, Markus and Pandey, Anurag},
  title =	{{Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{8:1--8:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.8},
  URN =		{urn:nbn:de:0030-drops-126112},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.8},
  annote =	{Keywords: Algebraic Complexity theory, Polynomial Identity Testing, Hitting Set, Pseudorandomness}
}
Document
DOI: 10.4230/LIPIcs.CCC.2020.21
Algebraic Branching Programs, Border Complexity, and Tangent Spaces

Authors: Markus Bläser, Christian Ikenmeyer, Meena Mahajan, Anurag Pandey, and Nitin Saurabh

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract
Nisan showed in 1991 that the width of a smallest noncommutative single-(source,sink) algebraic branching program (ABP) to compute a noncommutative polynomial is given by the ranks of specific matrices. This means that the set of noncommutative polynomials with ABP width complexity at most k is Zariski-closed, an important property in geometric complexity theory. It follows that approximations cannot help to reduce the required ABP width. It was mentioned by Forbes that this result would probably break when going from single-(source,sink) ABPs to trace ABPs. We prove that this is correct. Moreover, we study the commutative monotone setting and prove a result similar to Nisan, but concerning the analytic closure. We observe the same behavior here: The set of polynomials with ABP width complexity at most k is closed for single-(source,sink) ABPs and not closed for trace ABPs. The proofs reveal an intriguing connection between tangent spaces and the vector space of flows on the ABP. We close with additional observations on VQP and the closure of VNP which allows us to establish a separation between the two classes.
Cite as

Markus Bläser, Christian Ikenmeyer, Meena Mahajan, Anurag Pandey, and Nitin Saurabh. Algebraic Branching Programs, Border Complexity, and Tangent Spaces. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 21:1-21:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{blaser_et_al:LIPIcs.CCC.2020.21,
  author =	{Bl\"{a}ser, Markus and Ikenmeyer, Christian and Mahajan, Meena and Pandey, Anurag and Saurabh, Nitin},
  title =	{{Algebraic Branching Programs, Border Complexity, and Tangent Spaces}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{21:1--21:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.21},
  URN =		{urn:nbn:de:0030-drops-125733},
  doi =		{10.4230/LIPIcs.CCC.2020.21},
  annote =	{Keywords: Algebraic Branching Programs, Border Complexity, Tangent Spaces, Lower Bounds, Geometric Complexity Theory, Flows, VQP, VNP}
}
Document
Complete Volume
DOI: 10.4230/LIPIcs.STACS.2020
LIPIcs, Vol. 154, STACS 2020, Complete Volume

Authors: Christophe Paul and Markus Bläser

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Abstract
LIPIcs, Vol. 154, STACS 2020, Complete Volume
Cite as

37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 0-1024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@Proceedings{paul_et_al:LIPIcs.STACS.2020,
  title =	{{LIPIcs, Vol. 154, STACS 2020, Complete Volume}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020},
  URN =		{urn:nbn:de:0030-drops-118603},
  doi =		{10.4230/LIPIcs.STACS.2020},
  annote =	{Keywords: LIPIcs, Vol. 154, STACS 2020, Complete Volume}
}
Document
Front Matter
DOI: 10.4230/LIPIcs.STACS.2020.0
Front Matter, Table of Contents, Preface, Conference Organization

Authors: Christophe Paul and Markus Bläser

Published in: LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as

37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{paul_et_al:LIPIcs.STACS.2020.0,
  author =	{Paul, Christophe and Bl\"{a}ser, Markus},
  title =	{{Front Matter, Table of Contents, Preface, Conference Organization}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{0:i--0:xii},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Paul, Christophe and Bl\"{a}ser, Markus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.0},
  URN =		{urn:nbn:de:0030-drops-118614},
  doi =		{10.4230/LIPIcs.STACS.2020.0},
  annote =	{Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2019.3
Parameterized Valiant’s Classes

Authors: Markus Bläser and Christian Engels

Published in: LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

Abstract
We define a theory of parameterized algebraic complexity classes in analogy to parameterized Boolean counting classes. We define the classes VFPT and VW[t], which mirror the Boolean counting classes #FPT and #W[t], and define appropriate reductions and completeness notions. Our main contribution is the VW[1]-completeness proof of the parameterized clique family. This proof is far more complicated than in the Boolean world. It requires some new concepts like composition theorems for bounded exponential sums and Boolean-arithmetic formulas. In addition, we also look at two polynomials linked to the permanent with vastly different parameterized complexity.
Cite as

Markus Bläser and Christian Engels. Parameterized Valiant’s Classes. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 3:1-3:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{blaser_et_al:LIPIcs.IPEC.2019.3,
  author =	{Bl\"{a}ser, Markus and Engels, Christian},
  title =	{{Parameterized Valiant’s Classes}},
  booktitle =	{14th International Symposium on Parameterized and Exact Computation (IPEC 2019)},
  pages =	{3:1--3:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-129-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{148},
  editor =	{Jansen, Bart M. P. and Telle, Jan Arne},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.3},
  URN =		{urn:nbn:de:0030-drops-114648},
  doi =		{10.4230/LIPIcs.IPEC.2019.3},
  annote =	{Keywords: Algebraic complexity theory, parameterized complexity theory, Valiant’s classes}
}
Document
DOI: 10.4230/DagRep.8.9.133
Algebraic Methods in Computational Complexity (Dagstuhl Seminar 18391)

Authors: Markus Bläser, Valentine Kabanets, Jacobo Torán, and Christopher Umans

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

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.
Cite as

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-dev.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
DOI: 10.4230/LIPIcs.ITCS.2019.47
On the Complexity of Symmetric Polynomials

Authors: Markus Bläser and Gorav Jindal

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Abstract
The fundamental theorem of symmetric polynomials states that for a symmetric polynomial f_{Sym} in C[x_1,x_2,...,x_n], there exists a unique "witness" f in C[y_1,y_2,...,y_n] such that f_{Sym}=f(e_1,e_2,...,e_n), where the e_i's are the elementary symmetric polynomials. In this paper, we study the arithmetic complexity L(f) of the witness f as a function of the arithmetic complexity L(f_{Sym}) of f_{Sym}. We show that the arithmetic complexity L(f) of f is bounded by poly(L(f_{Sym}),deg(f),n). To the best of our knowledge, prior to this work only exponential upper bounds were known for L(f). The main ingredient in our result is an algebraic analogue of Newton's iteration on power series. As a corollary of this result, we show that if VP != VNP then there exist symmetric polynomial families which have super-polynomial arithmetic complexity. Furthermore, we study the complexity of testing whether a function is symmetric. For polynomials, this question is equivalent to arithmetic circuit identity testing. In contrast to this, we show that it is hard for Boolean functions.
Cite as

Markus Bläser and Gorav Jindal. On the Complexity of Symmetric Polynomials. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{blaser_et_al:LIPIcs.ITCS.2019.47,
  author =	{Bl\"{a}ser, Markus and Jindal, Gorav},
  title =	{{On the Complexity of Symmetric Polynomials}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{47:1--47:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.47},
  URN =		{urn:nbn:de:0030-drops-101402},
  doi =		{10.4230/LIPIcs.ITCS.2019.47},
  annote =	{Keywords: Symmetric Polynomials, Arithmetic Circuits, Arithmetic Complexity, Power Series, Elementary Symmetric Polynomials, Newton's Iteration}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2018.18
Graph Pattern Polynomials

Authors: Markus Bläser, Balagopal Komarath, and Karteek Sreenivasaiah

Published in: LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

Abstract
Given a host graph G and a pattern graph H, the induced subgraph isomorphism problem is to decide whether G contains an induced subgraph that is isomorphic to H. We study the time complexity of induced subgraph isomorphism problems when the pattern graph is fixed. Nesetril and Poljak gave an O(n^{k omega}) time algorithm that decides the induced subgraph isomorphism problem for any 3k vertex pattern graph (the universal algorithm), where omega is the matrix multiplication exponent. Improvements are not known for any infinite pattern family. Algorithms faster than the universal algorithm are known only for a finite number of pattern graphs. In this paper, we show that there exists infinitely many pattern graphs for which the induced subgraph isomorphism problem has algorithms faster than the universal algorithm. Our algorithm works by reducing the pattern detection problem into a multilinear term detection problem on special classes of polynomials called graph pattern polynomials. We show that many of the existing algorithms including the universal algorithm can also be described in terms of such a reduction. We formalize this class of algorithms by defining graph pattern polynomial families and defining a notion of reduction between these polynomial families. The reduction also allows us to argue about relative hardness of various graph pattern detection problems within this framework. We show that solving the induced subgraph isomorphism for any pattern graph that contains a k-clique is at least as hard detecting k-cliques. An equivalent theorem is not known in the general case. In the full version of this paper, we obtain new algorithms for P_5 and C_5 that are optimal under reasonable hardness assumptions. We also use this method to derive new combinatorial algorithms - algorithms that do not use fast matrix multiplication - for paths and cycles. We also show why graph homomorphisms play a major role in algorithms for subgraph isomorphism problems. Using this, we show that the arithmetic circuit complexity of the graph homomorphism polynomial for K_k - e (The k-clique with an edge removed) is related to the complexity of many subgraph isomorphism problems. This generalizes and unifies many existing results.
Cite as

Markus Bläser, Balagopal Komarath, and Karteek Sreenivasaiah. Graph Pattern Polynomials. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 18:1-18:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{blaser_et_al:LIPIcs.FSTTCS.2018.18,
  author =	{Bl\"{a}ser, Markus and Komarath, Balagopal and Sreenivasaiah, Karteek},
  title =	{{Graph Pattern Polynomials}},
  booktitle =	{38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)},
  pages =	{18:1--18:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-093-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{122},
  editor =	{Ganguly, Sumit and Pandya, Paritosh},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.18},
  URN =		{urn:nbn:de:0030-drops-99172},
  doi =		{10.4230/LIPIcs.FSTTCS.2018.18},
  annote =	{Keywords: algorithms, induced subgraph detection, algebraic framework}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2016.19
On Degeneration of Tensors and Algebras

Authors: Markus Bläser and Vladimir Lysikov

Published in: LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Abstract
An important building block in all current asymptotically fast algorithms for matrix multiplication are tensors with low border rank, that is, tensors whose border rank is equal or very close to their size. To find new asymptotically fast algorithms for matrix multiplication, it seems to be important to understand those tensors whose border rank is as small as possible, so called tensors of minimal border rank. We investigate the connection between degenerations of associative algebras and degenerations of their structure tensors in the sense of Strassen. It allows us to describe an open subset of n*n*n tensors of minimal border rank in terms of smoothability of commutative algebras. We describe the smoothable algebra associated to the Coppersmith-Winograd tensor and prove a lower bound for the border rank of the tensor used in the "easy construction" of Coppersmith and Winograd.
Cite as

Markus Bläser and Vladimir Lysikov. On Degeneration of Tensors and Algebras. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 19:1-19:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{blaser_et_al:LIPIcs.MFCS.2016.19,
  author =	{Bl\"{a}ser, Markus and Lysikov, Vladimir},
  title =	{{On Degeneration of Tensors and Algebras}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{19:1--19:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.19},
  URN =		{urn:nbn:de:0030-drops-64343},
  doi =		{10.4230/LIPIcs.MFCS.2016.19},
  annote =	{Keywords: bilinear complexity, border rank, commutative algebras, lower bounds}
}
Document
DOI: 10.4230/LIPIcs.STACS.2008.1337
On the Complexity of the Interlace Polynomial

Authors: Markus Bläser and Christian Hoffmann

Published in: LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

Abstract
We consider the two-variable interlace polynomial introduced by Arratia, Bollob`as and Sorkin (2004). We develop two graph transformations which allow us to derive point-to-point reductions for the interlace polynomial. Exploiting these reductions we obtain new results concerning the computational complexity of evaluating the interlace polynomial at a fixed point. Regarding exact evaluation, we prove that the interlace polynomial is #P-hard to evaluate at every point of the plane, except at one line, where it is trivially polynomial time computable, and four lines and two points, where the complexity mostly is still open. This solves a problem posed by Arratia, Bollob`as and Sorkin (2004). In particular, we observe that three specializations of the two-variable interlace polynomial, the vertex-nullity interlace polynomial, the vertex-rank interlace polynomial and the independent set polynomial, are almost everywhere #P-hard to evaluate, too. For the independent set polynomial, our reductions allow us to prove that it is even hard to approximate at every point except at $-1$ and~$0$.
Cite as

Markus Bläser and Christian Hoffmann. On the Complexity of the Interlace Polynomial. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 97-108, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{blaser_et_al:LIPIcs.STACS.2008.1337,
  author =	{Bl\"{a}ser, Markus and Hoffmann, Christian},
  title =	{{On the Complexity of the Interlace Polynomial}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{97--108},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Albers, Susanne and Weil, Pascal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1337},
  URN =		{urn:nbn:de:0030-drops-13378},
  doi =		{10.4230/LIPIcs.STACS.2008.1337},
  annote =	{Keywords: Computational complexity, approximation, interlace polynomial, independent set polynomial, graph transformation}
}
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