21 Search Results for "Bläser, Markus"

Volume

LIPIcs, Volume 187

38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

STACS 2021, March 16-19, 2021, Saarbrücken, Germany (Virtual Conference)

Editors: Markus Bläser and Benjamin Monmege

Volume

LIPIcs, Volume 154

37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

STACS 2020, March 10-13, 2020, Montpellier, France

Editors: Christophe Paul and Markus Bläser

Document
DOI: 10.4230/LIPIcs.CCC.2023.2
Border Complexity of Symbolic Determinant Under Rank One Restriction

Authors: Abhranil Chatterjee, Sumanta Ghosh, Rohit Gurjar, and Roshan Raj

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
VBP is the class of polynomial families that can be computed by the determinant of a symbolic matrix of the form A_0 + ∑_{i=1}^n A_i x_i where the size of each A_i is polynomial in the number of variables (equivalently, computable by polynomial-sized algebraic branching programs (ABP)). A major open problem in geometric complexity theory (GCT) is to determine whether VBP is closed under approximation i.e. whether VBP = VBP^ ̅. The power of approximation is well understood for some restricted models of computation, e.g. the class of depth-two circuits, read-once oblivious ABPs (ROABP), monotone ABPs, depth-three circuits of bounded top fan-in, and width-two ABPs. The former three classes are known to be closed under approximation [Markus Bläser et al., 2020], whereas the approximative closure of the last one captures the entire class of polynomial families computable by polynomial-sized formulas [Bringmann et al., 2017]. In this work, we consider the subclass of VBP computed by the determinant of a symbolic matrix of the form A_0 + ∑_{i=1}^n A_i x_i where for each 1 ≤ i ≤ n, A_i is of rank one. This class has been studied extensively [Edmonds, 1968; Jack Edmonds, 1979; Murota, 1993] and efficient identity testing algorithms are known for it [Lovász, 1989; Rohit Gurjar and Thomas Thierauf, 2020]. We show that this class is closed under approximation. In the language of algebraic geometry, we show that the set obtained by taking coordinatewise products of pairs of points from (the Plücker embedding of) a Grassmannian variety is closed.
Cite as

Abhranil Chatterjee, Sumanta Ghosh, Rohit Gurjar, and Roshan Raj. Border Complexity of Symbolic Determinant Under Rank One Restriction. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2023.2,
  author =	{Chatterjee, Abhranil and Ghosh, Sumanta and Gurjar, Rohit and Raj, Roshan},
  title =	{{Border Complexity of Symbolic Determinant Under Rank One Restriction}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{2:1--2:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.2},
  URN =		{urn:nbn:de:0030-drops-182721},
  doi =		{10.4230/LIPIcs.CCC.2023.2},
  annote =	{Keywords: Border Complexity, Symbolic Determinant, Valuated Matroid}
}
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
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.83
Monotone Arithmetic Complexity of Graph Homomorphism Polynomials

Authors: Balagopal Komarath, Anurag Pandey, and Chengot Sankaramenon Rahul

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph H to n-vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new algorithms for counting and detecting graph patterns, and also for obtaining natural polynomial families which are complete for algebraic complexity classes VBP, VP, and VNP. We discover that, in the monotone setting, the formula complexity, the ABP complexity, and the circuit complexity of such polynomial families are exactly characterized by the treedepth, the pathwidth, and the treewidth of the pattern graph respectively. Furthermore, we establish a single, unified framework, using our characterization, to collect several known results that were obtained independently via different methods. For instance, we attain superpolynomial separations between circuits, ABPs, and formulas in the monotone setting, where the polynomial families separating the classes all correspond to well-studied combinatorial problems. Moreover, our proofs rediscover fine-grained separations between these models for constant-degree polynomials.
Cite as

Balagopal Komarath, Anurag Pandey, and Chengot Sankaramenon Rahul. Monotone Arithmetic Complexity of Graph Homomorphism Polynomials. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 83:1-83:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{komarath_et_al:LIPIcs.ICALP.2022.83,
  author =	{Komarath, Balagopal and Pandey, Anurag and Rahul, Chengot Sankaramenon},
  title =	{{Monotone Arithmetic Complexity of Graph Homomorphism Polynomials}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{83:1--83:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.83},
  URN =		{urn:nbn:de:0030-drops-164245},
  doi =		{10.4230/LIPIcs.ICALP.2022.83},
  annote =	{Keywords: Homomorphism polynomials, Monotone complexity, Algebraic complexity, Graph algorithms, Fine-grained complexity, Fixed-parameter algorithms and complexity, Treewidth, Pathwidth, Treedepth, Graph homomorphisms, Algebraic circuits, Algebraic branching programs, Algebraic formulas}
}
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.
Cite as

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}
}
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