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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

We study the complexity of isomorphism problems for d-way arrays, or tensors, under natural actions by classical groups such as orthogonal, unitary, and symplectic groups. These problems arise naturally in statistical data analysis and quantum information. We study two types of complexity-theoretic questions. First, for a fixed action type (isomorphism, conjugacy, etc.), we relate the complexity of the isomorphism problem over a classical group to that over the general linear group. Second, for a fixed group type (orthogonal, unitary, or symplectic), we compare the complexity of the isomorphism problems for different actions.
Our main results are as follows. First, for orthogonal and symplectic groups acting on 3-way arrays, the isomorphism problems reduce to the corresponding problems over the general linear group. Second, for orthogonal and unitary groups, the isomorphism problems of five natural actions on 3-way arrays are polynomial-time equivalent, and the d-tensor isomorphism problem reduces to the 3-tensor isomorphism problem for any fixed d > 3. For unitary groups, the preceding result implies that LOCC classification of tripartite quantum states is at least as difficult as LOCC classification of d-partite quantum states for any d. Lastly, we also show that the graph isomorphism problem reduces to the tensor isomorphism problem over orthogonal and unitary groups.

Zhili Chen, Joshua A. Grochow, Youming Qiao, Gang Tang, and Chuanqi Zhang. On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials III: Actions by Classical Groups. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 31:1-31:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chen_et_al:LIPIcs.ITCS.2024.31, author = {Chen, Zhili and Grochow, Joshua A. and Qiao, Youming and Tang, Gang and Zhang, Chuanqi}, title = {{On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials III: Actions by Classical Groups}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {31:1--31:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.31}, URN = {urn:nbn:de:0030-drops-195595}, doi = {10.4230/LIPIcs.ITCS.2024.31}, annote = {Keywords: complexity class, tensor isomorphism, polynomial isomorphism, group isomorphism, local operations and classical communication} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. Sénizergues and the fifth author (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable in PSPACE when the input is given in terms of so-called virtually free presentations. Here we consider the isomorphism problem for the class of plain groups, that is, groups that are isomorphic to a free product of finitely many finite groups and finitely many copies of the infinite cyclic group. Every plain group is naturally and efficiently presented via an inverse-closed finite convergent length-reducing rewriting system. We prove that the isomorphism problem for plain groups given in this form lies in the polynomial time hierarchy, more precisely, in Σ₃^𝖯. This result is achieved by combining new geometric and algebraic characterisations of groups presented by inverse-closed finite convergent length-reducing rewriting systems developed in recent work of the second and third authors (2021) with classical finite group isomorphism results of Babai and Szemerédi (1984).

Heiko Dietrich, Murray Elder, Adam Piggott, Youming Qiao, and Armin Weiß. The Isomorphism Problem for Plain Groups Is in Σ₃^{𝖯}. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dietrich_et_al:LIPIcs.STACS.2022.26, author = {Dietrich, Heiko and Elder, Murray and Piggott, Adam and Qiao, Youming and Wei{\ss}, Armin}, title = {{The Isomorphism Problem for Plain Groups Is in \Sigma₃^\{𝖯\}}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {26:1--26:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.26}, URN = {urn:nbn:de:0030-drops-158368}, doi = {10.4230/LIPIcs.STACS.2022.26}, annote = {Keywords: plain group, isomorphism problem, polynomial hierarchy, \Sigma₃^\{𝖯\} complexity class, inverse-closed finite convergent length-reducing rewriting system} }

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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

One approach to make progress on the symbolic determinant identity testing (SDIT) problem is to study the structure of singular matrix spaces. After settling the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, Found. Comput. Math. 2020; Ivanyos-Qiao-Subrahmanyam, Comput. Complex. 2018), a natural next step is to understand singular matrix spaces whose non-commutative rank is full. At present, examples of such matrix spaces are mostly sporadic, so it is desirable to discover them in a more systematic way.
In this paper, we make a step towards this direction, by studying the family of matrix spaces that are closed under the commutator operation, that is, matrix Lie algebras. On the one hand, we demonstrate that matrix Lie algebras over the complex number field give rise to singular matrix spaces with full non-commutative ranks. On the other hand, we show that SDIT of such spaces can be decided in deterministic polynomial time. Moreover, we give a characterization for the matrix Lie algebras to yield a matrix space possessing singularity certificates as studied by Lovász (B. Braz. Math. Soc., 1989) and Raz and Wigderson (Building Bridges II, 2019).

Gábor Ivanyos, Tushant Mittal, and Youming Qiao. Symbolic Determinant Identity Testing and Non-Commutative Ranks of Matrix Lie Algebras. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 87:1-87:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ivanyos_et_al:LIPIcs.ITCS.2022.87, author = {Ivanyos, G\'{a}bor and Mittal, Tushant and Qiao, Youming}, title = {{Symbolic Determinant Identity Testing and Non-Commutative Ranks of Matrix Lie Algebras}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {87:1--87:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.87}, URN = {urn:nbn:de:0030-drops-156837}, doi = {10.4230/LIPIcs.ITCS.2022.87}, annote = {Keywords: derandomization, polynomial identity testing, symbolic determinant, non-commutative rank, Lie algebras} }

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

In this paper we study some classical complexity-theoretic questions regarding Group Isomorphism (GpI). We focus on p-groups (groups of prime power order) with odd p, which are believed to be a bottleneck case for GpI, and work in the model of matrix groups over finite fields. Our main results are as follows.
- Although search-to-decision and counting-to-decision reductions have been known for over four decades for Graph Isomorphism (GI), they had remained open for GpI, explicitly asked by Arvind & Torán (Bull. EATCS, 2005). Extending methods from Tensor Isomorphism (Grochow & Qiao, ITCS 2021), we show moderately exponential-time such reductions within p-groups of class 2 and exponent p.
- Despite the widely held belief that p-groups of class 2 and exponent p are the hardest cases of GpI, there was no reduction to these groups from any larger class of groups. Again using methods from Tensor Isomorphism (ibid.), we show the first such reduction, namely from isomorphism testing of p-groups of "small" class and exponent p to those of class two and exponent p.
For the first results, our main innovation is to develop linear-algebraic analogues of classical graph coloring gadgets, a key technique in studying the structural complexity of GI. Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets. The analysis of one of our gadgets relies on a classical result from group theory regarding random generation of classical groups (Kantor & Lubotzky, Geom. Dedicata, 1990). For the nilpotency class reduction, we combine a runtime analysis of the Lazard Correspondence with Tensor Isomorphism-completeness results (Grochow & Qiao, ibid.).

Joshua A. Grochow and Youming Qiao. On p-Group Isomorphism: Search-To-Decision, Counting-To-Decision, and Nilpotency Class Reductions via Tensors. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 16:1-16:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{grochow_et_al:LIPIcs.CCC.2021.16, author = {Grochow, Joshua A. and Qiao, Youming}, title = {{On p-Group Isomorphism: Search-To-Decision, Counting-To-Decision, and Nilpotency Class Reductions via Tensors}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {16:1--16:38}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.16}, URN = {urn:nbn:de:0030-drops-142905}, doi = {10.4230/LIPIcs.CCC.2021.16}, annote = {Keywords: group isomorphism, search-to-decision reduction, counting-to-decision reduction, nilpotent group isomorphism, p-group isomorphism, tensor isomorphism} }

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

We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms f, g ∈ 𝔽_q[x_1, … , x_n], and decides whether f and g are isomorphic in time q^O(n) for most f. This average-case setting has direct practical implications, having been studied in multivariate cryptography since the 1990s. Our second result concerns the complexity of testing equivalence of alternating trilinear forms. This problem is of interest in both mathematics and cryptography. We show that this problem is polynomial-time equivalent to testing equivalence of symmetric trilinear forms, by showing that they are both Tensor Isomorphism-complete (Grochow & Qiao, ITCS 2021), therefore is equivalent to testing isomorphism of cubic forms over most fields.

Joshua A. Grochow, Youming Qiao, and Gang Tang. Average-Case Algorithms for Testing Isomorphism of Polynomials, Algebras, and Multilinear Forms. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 38:1-38:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{grochow_et_al:LIPIcs.STACS.2021.38, author = {Grochow, Joshua A. and Qiao, Youming and Tang, Gang}, title = {{Average-Case Algorithms for Testing Isomorphism of Polynomials, Algebras, and Multilinear Forms}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {38:1--38:17}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.38}, URN = {urn:nbn:de:0030-drops-136836}, doi = {10.4230/LIPIcs.STACS.2021.38}, annote = {Keywords: polynomial isomorphism, trilinear form equivalence, algebra isomorphism, average-case algorithms, tensor isomorphism complete, symmetric and alternating bilinear maps} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

We study the complexity of isomorphism problems for tensors, groups, and polynomials. These problems have been studied in multivariate cryptography, machine learning, quantum information, and computational group theory. We show that these problems are all polynomial-time equivalent, creating bridges between problems traditionally studied in myriad research areas. This prompts us to define the complexity class TI, namely problems that reduce to the Tensor Isomorphism (TI) problem in polynomial time. Our main technical result is a polynomial-time reduction from d-tensor isomorphism to 3-tensor isomorphism. In the context of quantum information, this result gives multipartite-to-tripartite entanglement transformation procedure, that preserves equivalence under stochastic local operations and classical communication (SLOCC).

Joshua A. Grochow and Youming Qiao. On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 31:1-31:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{grochow_et_al:LIPIcs.ITCS.2021.31, author = {Grochow, Joshua A. and Qiao, Youming}, title = {{On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {31:1--31:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.31}, URN = {urn:nbn:de:0030-drops-135702}, doi = {10.4230/LIPIcs.ITCS.2021.31}, annote = {Keywords: complexity class, tensor isomorphism, polynomial isomorphism, group isomorphism, stochastic local operations and classical communication} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Motivated by testing isomorphism of p-groups, we study the alternating matrix space isometry problem (AltMatSpIso), which asks to decide whether two m-dimensional subspaces of n×n alternating (skew-symmetric if the field is not of characteristic 2) matrices are the same up to a change of basis. Over a finite field 𝔽_p with some prime p≠2, solving AltMatSpIso in time p^O(n+m) is equivalent to testing isomorphism of p-groups of class 2 and exponent p in time polynomial in the group order. The latter problem has long been considered a bottleneck case for the group isomorphism problem.
Recently, Li and Qiao presented an average-case algorithm for AltMatSpIso in time p^O(n) when n and m are linearly related (FOCS '17). In this paper, we present an average-case algorithm for AltMatSpIso in time p^O(n+m). Besides removing the restriction on the relation between n and m, our algorithm is considerably simpler, and the average-case analysis is stronger. We then implement our algorithm, with suitable modifications, in Magma. Our experiments indicate that it improves significantly over default (brute-force) algorithms for this problem.

Peter A. Brooksbank, Yinan Li, Youming Qiao, and James B. Wilson. Improved Algorithms for Alternating Matrix Space Isometry: From Theory to Practice. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{brooksbank_et_al:LIPIcs.ESA.2020.26, author = {Brooksbank, Peter A. and Li, Yinan and Qiao, Youming and Wilson, James B.}, title = {{Improved Algorithms for Alternating Matrix Space Isometry: From Theory to Practice}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {26:1--26:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.26}, URN = {urn:nbn:de:0030-drops-128920}, doi = {10.4230/LIPIcs.ESA.2020.26}, annote = {Keywords: Alternating Matrix Spaces, Average-case Algorithm, p-groups of Class 2nd Exponent p, Magma} }

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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

In the 1970’s, Lovász built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings (FCT 1979). A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, FOCS 2016; Ivanyos-Qiao-Subrahmanyam, ITCS 2017). In this paper, we lay the foundation for another bridge between graphs and alternating matrix spaces, in the context of independent sets and vertex colorings. The corresponding structures in alternating matrix spaces are isotropic spaces and isotropic decompositions, both useful structures in group theory and manifold theory.
We first show that the maximum independent set problem and the vertex c-coloring problem reduce to the maximum isotropic space problem and the isotropic c-decomposition problem, respectively. Next, we show that several topics and results about independent sets and vertex colorings have natural correspondences for isotropic spaces and decompositions. These include algorithmic problems, such as the maximum independent set problem for bipartite graphs, and exact exponential-time algorithms for the chromatic number, as well as mathematical questions, such as the number of maximal independent sets, and the relation between the maximum degree and the chromatic number. These connections lead to new interactions between graph theory and algebra. Some results have concrete applications to group theory and manifold theory, and we initiate a variant of these structures in the context of quantum information theory. Finally, we propose several open questions for further exploration.
(Dedicated to the memory of Ker-I Ko)

Xiaohui Bei, Shiteng Chen, Ji Guan, Youming Qiao, and Xiaoming Sun. From Independent Sets and Vertex Colorings to Isotropic Spaces and Isotropic Decompositions: Another Bridge Between Graphs and Alternating Matrix Spaces. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 8:1-8:48, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bei_et_al:LIPIcs.ITCS.2020.8, author = {Bei, Xiaohui and Chen, Shiteng and Guan, Ji and Qiao, Youming and Sun, Xiaoming}, title = {{From Independent Sets and Vertex Colorings to Isotropic Spaces and Isotropic Decompositions: Another Bridge Between Graphs and Alternating Matrix Spaces}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {8:1--8:48}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.8}, URN = {urn:nbn:de:0030-drops-116932}, doi = {10.4230/LIPIcs.ITCS.2020.8}, annote = {Keywords: independent set, vertex coloring, graphs, matrix spaces, isotropic subspace} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Let {\mathcal B} be a linear space of matrices over a field {\mathbb spanned by n\times n
matrices B_1, \dots, B_m. The non-commutative rank of {\mathcal B}$ is the minimum r\in {\mathbb N} such that there exists U\leq {\mathbb F}^n satisfying \dim(U)-\dim( {\mathcal B} (U))\geq
n-r, where {\mathcal B}(U):={\mathrm span}(\cup_{i\in[m]} B_i(U)).
Computing the non-commutative rank generalizes some well-known problems including the bipartite graph maximum
matching problem and the linear matroid intersection problem.
In this paper we give a deterministic polynomial-time algorithm to compute the
non-commutative rank over
any field {\mathbb F}. Prior to our work, such
an
algorithm was only known over the rational number field {\mathbb Q}, a result due to Garg et al, [GGOW]. Our algorithm is constructive and produces a witness
certifying the non-commutative rank, a feature that is missing in the algorithm from [GGOW].
Our result is built on techniques which we developed in a previous paper [IQS1], with a new reduction procedure that
helps to keep the blow-up parameter small. There are two ways to realize this
reduction. The first involves constructivizing a key result
of Derksen and Makam [DM2] which they developed in order to prove that the null cone
of matrix semi-invariants is cut out by generators whose degree is polynomial in the size of the matrices involved. We also give a second, simpler method to achieve this. This
gives another proof of the polynomial upper bound on the degree of the generators cutting out the null cone of matrix
semi-invariants.
Both the invariant-theoretic result and the algorithmic result rely crucially
on the regularity lemma proved in [IQS1]. In
this paper we improve on the constructive version of the regularity lemma from [IQS1] by removing a technical coprime
condition that was assumed there.

Gábor Ivanyos, Youming Qiao, and K Venkata Subrahmanyam. Constructive Non-Commutative Rank Computation Is in Deterministic Polynomial Time. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 55:1-55:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ivanyos_et_al:LIPIcs.ITCS.2017.55, author = {Ivanyos, G\'{a}bor and Qiao, Youming and Subrahmanyam, K Venkata}, title = {{Constructive Non-Commutative Rank Computation Is in Deterministic Polynomial Time}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {55:1--55:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.55}, URN = {urn:nbn:de:0030-drops-81667}, doi = {10.4230/LIPIcs.ITCS.2017.55}, annote = {Keywords: invariant theory, non-commutative rank, null cone, symbolic determinant identity testing, semi-invariants of quivers} }

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**Published in:** LIPIcs, Volume 79, 32nd Computational Complexity Conference (CCC 2017)

The complexity class PPA consists of NP-search problems which are reducible to the parity principle in undirected graphs. It contains a wide variety of interesting problems from graph theory, combinatorics, algebra and number theory, but only a few of these are known to be complete in the class. Before this work, the known complete problems were all discretizations or combinatorial analogues of topological fixed point theorems.
Here we prove the PPA-completeness of two problems of radically different style. They are PPA-Circuit CNSS and PPA-Circuit Chevalley, related respectively to the Combinatorial Nullstellensatz and to the Chevalley-Warning Theorem over the two elements field GF(2). The input of these problems contain PPA-circuits which are arithmetic circuits with special symmetric properties that assure that the polynomials computed by them have always an even number of zeros. In the proof of the result we relate the multilinear degree of the polynomials to the parity of the maximal parse subcircuits that compute monomials with maximal multilinear degree, and we show that the maximal parse subcircuits of a PPA-circuit can be paired in polynomial time.

Aleksandrs Belovs, Gábor Ivanyos, Youming Qiao, Miklos Santha, and Siyi Yang. On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz. In 32nd Computational Complexity Conference (CCC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 79, pp. 30:1-30:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{belovs_et_al:LIPIcs.CCC.2017.30, author = {Belovs, Aleksandrs and Ivanyos, G\'{a}bor and Qiao, Youming and Santha, Miklos and Yang, Siyi}, title = {{On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz}}, booktitle = {32nd Computational Complexity Conference (CCC 2017)}, pages = {30:1--30:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-040-8}, ISSN = {1868-8969}, year = {2017}, volume = {79}, editor = {O'Donnell, Ryan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2017.30}, URN = {urn:nbn:de:0030-drops-75260}, doi = {10.4230/LIPIcs.CCC.2017.30}, annote = {Keywords: Chevalley-Warning Theorem, Combinatorail Nullstellensatz, Polynomial Parity Argument, arithmetic circuit} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

One fundamental question in the context of the geometric complexity theory approach to the VP vs. VNP conjecture is whether VP = !VP, where VP is the class of families of polynomials that can be computed by arithmetic circuits of polynomial degree and size, and VP is the class of families of polynomials that can be approximated infinitesimally closely by arithmetic circuits of polynomial degree and size. The goal of this article is to study the conjecture in (Mulmuley, FOCS 2012) that !VP is not contained in VP.
Towards that end, we introduce three degenerations of VP (i.e., sets of points in VP), namely the stable degeneration Stable-VP, the Newton degeneration Newton-VP, and the p-definable one-parameter degeneration VP*. We also introduce analogous degenerations of VNP. We show that Stable-VP subseteq Newton-VP subseteq VP* subseteq VNP, and Stable-VNP = Newton-VNP = VNP* = VNP. The three notions of degenerations and the proof of this result shed light on the problem of separating VP from VP.
Although we do not yet construct explicit candidates for the polynomial families in !VP\VP, we prove results which tell us where not to look for such families. Specifically, we demonstrate that the families in Newton-VP \VP based on semi-invariants of quivers would have to be nongeneric by showing that, for many finite quivers (including some wild ones), Newton degeneration of any generic semi-invariant can be computed by a circuit of polynomial size. We also show that the Newton degenerations of perfect matching Pfaffians, monotone arithmetic circuits over the reals, and Schur polynomials have polynomial-size circuits.

Joshua A. Grochow, Ketan D. Mulmuley, and Youming Qiao. Boundaries of VP and VNP. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{grochow_et_al:LIPIcs.ICALP.2016.34, author = {Grochow, Joshua A. and Mulmuley, Ketan D. and Qiao, Youming}, title = {{Boundaries of VP and VNP}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {34:1--34:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.34}, URN = {urn:nbn:de:0030-drops-63147}, doi = {10.4230/LIPIcs.ICALP.2016.34}, annote = {Keywords: geometric complexity theory, arithmetic circuit, border complexity} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

We design two deterministic polynomial time algorithms for variants of a problem introduced by Edmonds in 1967: determine the rank of a matrix M whose entries are homogeneous linear polynomials over the integers. Given a linear subspace B of the nxn matrices over some field F, we consider the following problems: symbolic matrix rank (SMR) is the problem to determine the maximum rank among matrices in B, while symbolic determinant identity testing (SDIT) is the question to decide whether there exists a nonsingular matrix in B. The constructive versions of these problems are asking to find a matrix of maximum rank, respectively a nonsingular matrix, if there exists one.
Our first algorithm solves the constructive SMR when B is spanned by unknown rank one matrices, answering an open question of Gurvits. Our second algorithm solves the constructive SDIT when B is spanned by triangularizable matrices, but the triangularization is not given explicitly. Both algorithms work over finite fields of size at least n+1 and over the rational numbers, and the first algorithm actually solves (the non-constructive) SMR independent of the field size. Our main tool to obtain these results is to generalize Wong sequences, a classical method to deal with pairs of matrices, to the case of pairs of matrix spaces.

Gábor Ivanyos, Marek Karpinski, Youming Qiao, and Miklos Santha. Generalized Wong sequences and their applications to Edmonds' problems. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 397-408, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{ivanyos_et_al:LIPIcs.STACS.2014.397, author = {Ivanyos, G\'{a}bor and Karpinski, Marek and Qiao, Youming and Santha, Miklos}, title = {{Generalized Wong sequences and their applications to Edmonds' problems}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {397--408}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.397}, URN = {urn:nbn:de:0030-drops-44741}, doi = {10.4230/LIPIcs.STACS.2014.397}, annote = {Keywords: symbolic determinantal identity testing, Edmonds' problem, maximum rank matrix completion, derandomization, Wong sequences} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

We consider the problem of testing isomorphism of groups of order n
given by Cayley tables. The trivial n^{log n} bound on the time
complexity for the general case has not been improved over the past
four decades. Recently, Babai et al. (following Babai et al. in SODA
2011) presented a polynomial-time algorithm for groups without abelian
normal subgroups, which suggests solvable groups as the hard case for
group isomorphism problem. Extending recent work by Le Gall (STACS
2009) and Qiao et al. (STACS 2011), in this paper we design a
polynomial-time algorithm to test isomorphism for the largest class of
solvable groups yet, namely groups with abelian Sylow towers, defined
as follows. A group G is said to possess a Sylow tower, if there
exists a normal series where each quotient is isomorphic to Sylow
subgroup of G. A group has an abelian Sylow tower if it has a Sylow
tower and all its Sylow subgroups are abelian. In fact, we are able
to compute the coset of isomorphisms of groups formed as coprime
extensions of an abelian group, by a group whose automorphism group is
known.
The mathematical tools required include representation theory,
Wedderburn's theorem on semisimple algebras, and M.E. Harris's 1980
work on p'-automorphisms of abelian p-groups. We use tools from the
theory of permutation group algorithms, and develop an algorithm for a
parameterized versin of the graph-isomorphism-hard setwise stabilizer
problem, which may be of independent interest.

László Babai and Youming Qiao. Polynomial-time Isomorphism Test for Groups with Abelian Sylow Towers. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 453-464, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{babai_et_al:LIPIcs.STACS.2012.453, author = {Babai, L\'{a}szl\'{o} and Qiao, Youming}, title = {{Polynomial-time Isomorphism Test for Groups with Abelian Sylow Towers}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {453--464}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.453}, URN = {urn:nbn:de:0030-drops-34008}, doi = {10.4230/LIPIcs.STACS.2012.453}, annote = {Keywords: polynomial-time algorithm, group isomorphism, solvable group} }

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**Published in:** LIPIcs, Volume 9, 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)

A normal Hall subgroup $N$ of a group $G$ is a normal subgroup with its order coprime with its index. Schur-Zassenhaus theorem states that every normal Hall subgroup has a complement subgroup, that is a set of coset representatives H which also forms a subgroup of G. In this paper, we present a framework to test isomorphism of groups with at least one normal Hall subgroup, when groups are given as multiplication tables. To establish the framework, we first observe that a proof of Schur-Zassenhaus theorem is constructive, and formulate a necessary and sufficient condition for testing isomorphism in terms of the associated actions of the semidirect products, and isomorphisms of the normal parts and complement parts.
We then focus on the case when the normal subgroup is abelian. Utilizing basic facts of representation theory of finite groups and a technique by Le Gall [STACS 2009], we first get an efficient isomorphism testing algorithm when the complement has bounded number of generators. For the case when the complement subgroup is elementary abelian, which does not necessarily have bounded number of generators, we obtain a polynomial time isomorphism testing algorithm by reducing to generalized code isomorphism problem. A solution to the latter can be obtained by a mild extension of the singly exponential (in the number of coordinates) time algorithm for code isomorphism problem developed recently by Babai. Enroute to obtaining the above reduction, we study the following computational problem in representation theory of finite groups: given two representations rho and tau of a group $H$ over Z_p^d , p a prime, determine if there exists an automorphism phi:H -> H, such that the induced representation rho_phi=rho o phi and tau are equivalent, in time poly(|H|,p^d).

Youming Qiao, Jayalal Sarma M.N., and Bangsheng Tang. On Isomorphism Testing of Groups with Normal Hall Subgroups. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 9, pp. 567-578, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{qiao_et_al:LIPIcs.STACS.2011.567, author = {Qiao, Youming and Sarma M.N., Jayalal and Tang, Bangsheng}, title = {{On Isomorphism Testing of Groups with Normal Hall Subgroups}}, booktitle = {28th International Symposium on Theoretical Aspects of Computer Science (STACS 2011)}, pages = {567--578}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-25-5}, ISSN = {1868-8969}, year = {2011}, volume = {9}, editor = {Schwentick, Thomas and D\"{u}rr, Christoph}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2011.567}, URN = {urn:nbn:de:0030-drops-30443}, doi = {10.4230/LIPIcs.STACS.2011.567}, annote = {Keywords: Group Isomorphism Problem, Normal Hall Subgroups, Computational Complexity} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

In this paper we study algebraic branching programs (ABPs) with restrictions on the order and the number of reads of variables in the program. An ABP is given by a layered directed acyclic graph with source $s$ and sink $t$, whose edges are labeled by variables taken from the set $\{x_1, x_2, \ldots, x_n\}$ or field constants. It computes the sum of weights of all paths from $s$ to $t$, where the weight of a path is defined as the product of edge-labels on the path.
Given a permutation $\pi$ of the $n$ variables, for a $\pi$-ordered ABP ($\pi$-OABP), for any directed path $p$ from $s$ to $t$, a variable can appear at most once on $p$, and the order in which variables appear on $p$ must respect $\pi$. One can think of OABPs as being the arithmetic analogue of ordered binary decision diagrams (OBDDs). We say an ABP $A$ is of read $r$, if any variable appears at most $r$ times in $A$.
Our main result pertains to the polynomial identity testing problem, i.e. the problem of deciding whether a given $n$-variate polynomial is identical to the zero polynomial or not. We prove that over any field $\F$, and in the black-box model, i.e. given only query access to the polynomial, read $r$ $\pi$-OABP computable polynomials can be tested in $\DTIME[2^{O(r\log r \cdot \log^2 n \log\log n)}]$. In case $\F$ is a finite field, the above time bound holds provided the identity testing algorithm is allowed to make queries to extension fields of $\F$. To establish this result, we combine some basic tools from algebraic geometry with ideas from derandomization in the Boolean domain.
Our next set of results investigates the computational limitations of OABPs. It is shown that any OABP computing the determinant or permanent requires size $\Omega(2^n/n)$ and read $\Omega(2^n/n^2)$. We give a multilinear polynomial $p$ in $2n+1$ variables over some specifically selected field $\mathbb{G}$, such that any OABP computing $p$ must read some variable at least $2^n$ times. We prove a strict separation for the computational power of read $(r-1)$ and read $r$ OABPs. Namely, we show that the elementary symmetric polynomial of degree $r$ in $n$ variables can be computed by a size $O(rn)$ read $r$ OABP, but not by a read $(r-1)$ OABP, for any $0 < 2r-1 \leq n$. Finally, we give an example of a polynomial $p$ and two variables orders $\pi \neq \pi'$, such that $p$ can be computed by a read-once $\pi$-OABP, but where any $\pi'$-OABP computing $p$ must read some variable at least $2^n$ times.

Maurice Jansen, Youming Qiao, and Jayalal Sarma M.N.. Deterministic Black-Box Identity Testing $pi$-Ordered Algebraic Branching Programs. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 296-307, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{jansen_et_al:LIPIcs.FSTTCS.2010.296, author = {Jansen, Maurice and Qiao, Youming and Sarma M.N., Jayalal}, title = {{Deterministic Black-Box Identity Testing \$pi\$-Ordered Algebraic Branching Programs}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {296--307}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.296}, URN = {urn:nbn:de:0030-drops-28728}, doi = {10.4230/LIPIcs.FSTTCS.2010.296}, annote = {Keywords: ordered algebraic branching program, polynomial identity testing} }

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