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**Published in:** Dagstuhl Seminar Reports. Dagstuhl Seminar Reports, Volume 1 (2021)

D. K. Arvind, Kemal Ebcioglu, Christian Lengauer, Keshav Pingali, and Robert S. Schreiber. Instruction-Level Parallelism and Parallelizing Compilation (Dagstuhl Seminar 99161). Dagstuhl Seminar Report 237, pp. 1-30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (1999)

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@TechReport{arvind_et_al:DagSemRep.237, author = {Arvind, D. K. and Ebcioglu, Kemal and Lengauer, Christian and Pingali, Keshav and Schreiber, Robert S.}, title = {{Instruction-Level Parallelism and Parallelizing Compilation (Dagstuhl Seminar 99161)}}, pages = {1--30}, ISSN = {1619-0203}, year = {1999}, type = {Dagstuhl Seminar Report}, number = {237}, institution = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemRep.237}, URN = {urn:nbn:de:0030-drops-151237}, doi = {10.4230/DagSemRep.237}, }

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**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)

We consider a hierarchy of graph invariants that naturally extends the spectral invariants defined by Fürer (Lin. Alg. Appl. 2010) based on the angles formed by the set of standard basis vectors and their projections onto eigenspaces of the adjacency matrix. We provide a purely combinatorial characterization of this hierarchy in terms of the walk counts. This allows us to give a complete answer to Fürer’s question about the strength of his invariants in distinguishing non-isomorphic graphs in comparison to the 2-dimensional Weisfeiler-Leman algorithm, extending the recent work of Rattan and Seppelt (SODA 2023). As another application of the characterization, we prove that almost all graphs are determined up to isomorphism in terms of the spectrum and the angles, which is of interest in view of the long-standing open problem whether almost all graphs are determined by their eigenvalues alone. Finally, we describe the exact relationship between the hierarchy and the Weisfeiler-Leman algorithms for small dimensions, as also some other important spectral characteristics of a graph such as the generalized and the main spectra.

V. Arvind, Frank Fuhlbrück, Johannes Köbler, and Oleg Verbitsky. On a Hierarchy of Spectral Invariants for Graphs. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{arvind_et_al:LIPIcs.STACS.2024.6, author = {Arvind, V. and Fuhlbr\"{u}ck, Frank and K\"{o}bler, Johannes and Verbitsky, Oleg}, title = {{On a Hierarchy of Spectral Invariants for Graphs}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {6:1--6:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.6}, URN = {urn:nbn:de:0030-drops-197166}, doi = {10.4230/LIPIcs.STACS.2024.6}, annote = {Keywords: Graph Isomorphism, spectra of graphs, combinatorial refinement, strongly regular graphs} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

The identity testing of rational formulas (RIT) in the free skew field efficiently reduces to computing the rank of a matrix whose entries are linear polynomials in noncommuting variables [Hrubeš and Wigderson, 2015]. This rank computation problem has deterministic polynomial-time white-box algorithms [Ankit Garg et al., 2016; Ivanyos et al., 2018] and a randomized polynomial-time algorithm in the black-box setting [Harm Derksen and Visu Makam, 2017]. In this paper, we propose a new approach for efficient derandomization of black-box RIT. Additionally, we obtain results for matrix rank computation over the free skew field and construct efficient linear pencil representations for a new class of rational expressions. More precisely, we show:
- Under the hardness assumption that the ABP (algebraic branching program) complexity of every polynomial identity for the k×k matrix algebra is 2^Ω(k) [Andrej Bogdanov and Hoeteck Wee, 2005], we obtain a subexponential-time black-box RIT algorithm for rational formulas of inversion height almost logarithmic in the size of the formula. This can be seen as the first "hardness implies derandomization" type theorem for rational formulas.
- We show that the noncommutative rank of any matrix over the free skew field whose entries have small linear pencil representations can be computed in deterministic polynomial time. While an efficient rank computation was known for matrices with noncommutative formulas as entries [Ankit Garg et al., 2020], we obtain the first deterministic polynomial-time algorithms for rank computation of matrices whose entries are noncommutative ABPs or rational formulas.
- Motivated by the definition given by Bergman [George M Bergman, 1976], we define a new class of rational functions where a rational function of inversion height at most h is defined as a composition of a noncommutative r-skewed circuit (equivalently an ABP) with inverses of rational functions of this class of inversion height at most h-1 which are also disjoint. We obtain a polynomial-size linear pencil representation for this class which gives a white-box deterministic polynomial-time identity testing algorithm for the class.

V. Arvind, Abhranil Chatterjee, Utsab Ghosal, Partha Mukhopadhyay, and C. Ramya. On Identity Testing and Noncommutative Rank Computation over the Free Skew Field. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 6:1-6:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arvind_et_al:LIPIcs.ITCS.2023.6, author = {Arvind, V. and Chatterjee, Abhranil and Ghosal, Utsab and Mukhopadhyay, Partha and Ramya, C.}, title = {{On Identity Testing and Noncommutative Rank Computation over the Free Skew Field}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {6:1--6:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.6}, URN = {urn:nbn:de:0030-drops-175093}, doi = {10.4230/LIPIcs.ITCS.2023.6}, annote = {Keywords: Algebraic Complexity, Identity Testing, Non-commutative rank} }

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RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Hrubeš and Wigderson [Hrubeš and Wigderson, 2015] initiated the complexity-theoretic study of noncommutative formulas with inverse gates. They introduced the Rational Identity Testing (RIT) problem which is to decide whether a noncommutative rational formula computes zero in the free skew field. In the white-box setting, there are deterministic polynomial-time algorithms due to Garg, Gurvits, Oliveira, and Wigderson [Ankit Garg et al., 2016] and Ivanyos, Qiao, and Subrahmanyam [Ivanyos et al., 2018].
A central open problem in this area is to design an efficient deterministic black-box identity testing algorithm for rational formulas. In this paper, we solve this for the first nested inverse case. More precisely, we obtain a deterministic quasipolynomial-time black-box RIT algorithm for noncommutative rational formulas of inversion height two via a hitting set construction. Several new technical ideas are involved in the hitting set construction, including concepts from matrix coefficient realization theory [Volčič, 2018] and properties of cyclic division algebras [T.Y. Lam, 2001]. En route to the proof, an important step is to embed the hitting set of Forbes and Shpilka for noncommutative formulas [Michael A. Forbes and Amir Shpilka, 2013] inside a cyclic division algebra of small index.

V. Arvind, Abhranil Chatterjee, and Partha Mukhopadhyay. Black-Box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial Time. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 23:1-23:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{arvind_et_al:LIPIcs.APPROX/RANDOM.2022.23, author = {Arvind, V. and Chatterjee, Abhranil and Mukhopadhyay, Partha}, title = {{Black-Box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial Time}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {23:1--23:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.23}, URN = {urn:nbn:de:0030-drops-171451}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.23}, annote = {Keywords: Rational Identity Testing, Black-box Derandomization, Cyclic Division Algebra, Matrix coefficient realization theory} }

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**Published in:** LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Motivated by equivalence testing of k-tape automata, we study the equivalence testing of weighted automata in the more general setting, over partially commutative monoids (in short, pc monoids), and show efficient algorithms in some special cases, exploiting the structure of the underlying non-commutation graph of the monoid.
Specifically, if the edge clique cover number of the non-commutation graph of the pc monoid is a constant, we obtain a deterministic quasi-polynomial time algorithm for equivalence testing. As a corollary, we obtain the first deterministic quasi-polynomial time algorithms for equivalence testing of k-tape weighted automata and for equivalence testing of deterministic k-tape automata for constant k. Prior to this, the best complexity upper bound for these k-tape automata problems were randomized polynomial-time, shown by Worrell [James Worrell, 2013]. Finding a polynomial-time deterministic algorithm for equivalence testing of deterministic k-tape automata for constant k has been open for several years [Emily P. Friedman and Sheila A. Greibach, 1982] and our results make progress.
We also consider pc monoids for which the non-commutation graphs have an edge cover consisting of at most k cliques and star graphs for any constant k. We obtain a randomized polynomial-time algorithm for equivalence testing of weighted automata over such monoids.
Our results are obtained by designing efficient zero-testing algorithms for weighted automata over such pc monoids.

V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. Equivalence Testing of Weighted Automata over Partially Commutative Monoids. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arvind_et_al:LIPIcs.MFCS.2021.10, author = {Arvind, V. and Chatterjee, Abhranil and Datta, Rajit and Mukhopadhyay, Partha}, title = {{Equivalence Testing of Weighted Automata over Partially Commutative Monoids}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {10:1--10:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.10}, URN = {urn:nbn:de:0030-drops-144503}, doi = {10.4230/LIPIcs.MFCS.2021.10}, annote = {Keywords: Weighted Automata, Automata Equivalence, Partially Commutative Monoid} }

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

We explore a special case of rational identity testing and algorithmic versions of two theorems on noncommutative polynomials, namely, Amitsur's theorem [S.A Amitsur, 1966] and the Brešar-Klep theorem [Brešar and Klep, 2008] when the input polynomial is given by an algebraic branching program (ABP). Let f be a degree-d n-variate noncommutative polynomial in the free ring Q<x_1,x_2,...,x_n> over rationals.
1) We consider the following special case of rational identity testing: Given a noncommutative ABP as white-box, whose edge labels are linear forms or inverses of linear forms, we show a deterministic polynomial-time algorithm to decide if the rational function computed by it is equivalent to zero in the free skew field Q<(X)>. Given black-box access to the ABP, we give a deterministic quasi-polynomial time algorithm for this problem.
2) Amitsur's theorem implies that if a noncommutative polynomial f is nonzero on k x k matrices then, in fact, f(M_1,M_2,...,M_n) is invertible for some matrix tuple (M_1,M_2,...,M_n) in (M_k(ℚ))^n. While a randomized polynomial time algorithm to find such (M_1,M_2,...,M_n) given black-box access to f is simple, we obtain a deterministic s^{O(log d)} time algorithm for the problem with black-box access to f, where s is the minimum ABP size for f and d is the degree of f.
3) The Brešar-Klep Theorem states that the span of the range of any noncommutative polynomial f on k x k matrices over Q is one of the following: zero, scalar multiples of I_k, trace-zero matrices in M_k(Q), or all of M_k(Q). We obtain a deterministic polynomial-time algorithm to decide which case occurs, given white-box access to an ABP for f. We also give a deterministic s^{O(log d)} time algorithm given black-box access to an ABP of size s for f. Our algorithms work when k >= d.
Our techniques are based on some automata theory combined with known techniques for noncommutative ABP identity testing [Ran Raz and Amir Shpilka, 2005; Michael A. Forbes and Amir Shpilka, 2013].

V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. A Special Case of Rational Identity Testing and the Brešar-Klep Theorem. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 10:1-10:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{arvind_et_al:LIPIcs.MFCS.2020.10, author = {Arvind, V. and Chatterjee, Abhranil and Datta, Rajit and Mukhopadhyay, Partha}, title = {{A Special Case of Rational Identity Testing and the Bre\v{s}ar-Klep Theorem}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {10:1--10:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.10}, URN = {urn:nbn:de:0030-drops-126807}, doi = {10.4230/LIPIcs.MFCS.2020.10}, annote = {Keywords: Rational identity testing, ABP with inverses, Bre\v{s}ar-Klep Theorem, Invertible image, Amitsur’s theorem} }

Document

**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

Let C be an arithmetic circuit of poly(n) size given as input that computes a polynomial f in F[X], where X={x_1,x_2,...,x_n} and F is any field where the field arithmetic can be performed efficiently. We obtain new algorithms for the following two problems first studied by Koutis and Williams [Ioannis Koutis, 2008; Ryan Williams, 2009; Ioannis Koutis and Ryan Williams, 2016].
- (k,n)-MLC: Compute the sum of the coefficients of all degree-k multilinear monomials in the polynomial f.
- k-MMD: Test if there is a nonzero degree-k multilinear monomial in the polynomial f.
Our algorithms are based on the fact that the Hadamard product f o S_{n,k}, is the degree-k multilinear part of f, where S_{n,k} is the k^{th} elementary symmetric polynomial.
- For (k,n)-MLC problem, we give a deterministic algorithm of run time O^*(n^(k/2+c log k)) (where c is a constant), answering an open question of Koutis and Williams [Ioannis Koutis and Ryan Williams, 2016]. As corollaries, we show O^*(binom{n}{downarrow k/2})-time exact counting algorithms for several combinatorial problems: k-Tree, t-Dominating Set, m-Dimensional k-Matching.
- For k-MMD problem, we give a randomized algorithm of run time 4.32^k * poly(n,k). Our algorithm uses only poly(n,k) space. This matches the run time of a recent algorithm [Cornelius Brand et al., 2018] for k-MMD which requires exponential (in k) space.
Other results include fast deterministic algorithms for (k,n)-MLC and k-MMD problems for depth three circuits.

V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. Fast Exact Algorithms Using Hadamard Product of Polynomials. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{arvind_et_al:LIPIcs.FSTTCS.2019.9, author = {Arvind, V. and Chatterjee, Abhranil and Datta, Rajit and Mukhopadhyay, Partha}, title = {{Fast Exact Algorithms Using Hadamard Product of Polynomials}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {9:1--9:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.9}, URN = {urn:nbn:de:0030-drops-115711}, doi = {10.4230/LIPIcs.FSTTCS.2019.9}, annote = {Keywords: Hadamard Product, Multilinear Monomial Detection and Counting, Rectangular Permanent, Symmetric Polynomial} }

Document

**Published in:** LIPIcs, Volume 149, 30th International Symposium on Algorithms and Computation (ISAAC 2019)

We study the arithmetic circuit complexity of some well-known family of polynomials through the lens of parameterized complexity. Our main focus is on the construction of explicit algebraic branching programs (ABP) for determinant and permanent polynomials of the rectangular symbolic matrix in both commutative and noncommutative settings. The main results are:
- We show an explicit O^*(binom{n}{downarrow k/2})-size ABP construction for noncommutative permanent polynomial of k x n symbolic matrix. We obtain this via an explicit ABP construction of size O^*(binom{n}{downarrow k/2}) for S_{n,k}^*, noncommutative symmetrized version of the elementary symmetric polynomial S_{n,k}.
- We obtain an explicit O^*(2^k)-size ABP construction for the commutative rectangular determinant polynomial of the k x n symbolic matrix.
- In contrast, we show that evaluating the rectangular noncommutative determinant over rational matrices is #W[1]-hard.

V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. On Explicit Branching Programs for the Rectangular Determinant and Permanent Polynomials. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 38:1-38:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{arvind_et_al:LIPIcs.ISAAC.2019.38, author = {Arvind, V. and Chatterjee, Abhranil and Datta, Rajit and Mukhopadhyay, Partha}, title = {{On Explicit Branching Programs for the Rectangular Determinant and Permanent Polynomials}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {38:1--38:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.38}, URN = {urn:nbn:de:0030-drops-115340}, doi = {10.4230/LIPIcs.ISAAC.2019.38}, annote = {Keywords: Determinant, Permanent, Parameterized Complexity, Branching Programs} }

Document

RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Hrubeš and Wigderson [Pavel Hrubeš and Avi Wigderson, 2014] initiated the study of noncommutative arithmetic circuits with division computing a noncommutative rational function in the free skew field, and raised the question of rational identity testing. For noncommutative formulas with inverses the problem can be solved in deterministic polynomial time in the white-box model [Ankit Garg et al., 2016; Ivanyos et al., 2018]. It can be solved in randomized polynomial time in the black-box model [Harm Derksen and Visu Makam, 2017], where the running time is polynomial in the size of the formula. The complexity of identity testing of noncommutative rational functions, in general, remains open for noncommutative circuits with inverses.
We solve the problem for a natural special case. We consider expressions in the free group algebra F(X,X^{-1}) where X={x_1, x_2, ..., x_n}. Our main results are the following.
1) Given a degree d expression f in F(X,X^{-1}) as a black-box, we obtain a randomized poly(n,d) algorithm to check whether f is an identically zero expression or not. The technical contribution is an Amitsur-Levitzki type theorem [A. S. Amitsur and J. Levitzki, 1950] for F(X, X^{-1}). This also yields a deterministic identity testing algorithm (and even an expression reconstruction algorithm) that is polynomial time in the sparsity of the input expression.
2) Given an expression f in F(X,X^{-1}) of degree D and sparsity s, as black-box, we can check whether f is identically zero or not in randomized poly(n,log s, log D) time. This yields a randomized polynomial-time algorithm when D and s are exponential in n.

V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. Efficient Black-Box Identity Testing for Free Group Algebras. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 57:1-57:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{arvind_et_al:LIPIcs.APPROX-RANDOM.2019.57, author = {Arvind, V. and Chatterjee, Abhranil and Datta, Rajit and Mukhopadhyay, Partha}, title = {{Efficient Black-Box Identity Testing for Free Group Algebras}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {57:1--57:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.57}, URN = {urn:nbn:de:0030-drops-112723}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.57}, annote = {Keywords: Rational identity testing, Free group algebra, Noncommutative computation, Randomized algorithms} }

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

Let F[X] be the polynomial ring over the variables X={x_1,x_2, ..., x_n}. An ideal I= <p_1(x_1), ..., p_n(x_n)> generated by univariate polynomials {p_i(x_i)}_{i=1}^n is a univariate ideal. We study the ideal membership problem for the univariate ideals and show the following results.
- Let f(X) in F[l_1, ..., l_r] be a (low rank) polynomial given by an arithmetic circuit where l_i : 1 <= i <= r are linear forms, and I=<p_1(x_1), ..., p_n(x_n)> be a univariate ideal. Given alpha in F^n, the (unique) remainder f(X) mod I can be evaluated at alpha in deterministic time d^{O(r)} * poly(n), where d=max {deg(f),deg(p_1)...,deg(p_n)}. This yields a randomized n^{O(r)} algorithm for minimum vertex cover in graphs with rank-r adjacency matrices. It also yields an n^{O(r)} algorithm for evaluating the permanent of a n x n matrix of rank r, over any field F. Over Q, an algorithm of similar run time for low rank permanent is due to Barvinok [Barvinok, 1996] via a different technique.
- Let f(X)in F[X] be given by an arithmetic circuit of degree k (k treated as fixed parameter) and I=<p_1(x_1), ..., p_n(x_n)>. We show that in the special case when I=<x_1^{e_1}, ..., x_n^{e_n}>, we obtain a randomized O^*(4.08^k) algorithm that uses poly(n,k) space.
- Given f(X)in F[X] by an arithmetic circuit and I=<p_1(x_1), ..., p_k(x_k)>, membership testing is W[1]-hard, parameterized by k. The problem is MINI[1]-hard in the special case when I=<x_1^{e_1}, ..., x_k^{e_k}>.

V. Arvind, Abhranil Chatterjee, Rajit Datta, and Partha Mukhopadhyay. Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators. 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. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{arvind_et_al:LIPIcs.FSTTCS.2018.7, author = {Arvind, V. and Chatterjee, Abhranil and Datta, Rajit and Mukhopadhyay, Partha}, title = {{Univariate Ideal Membership Parameterized by Rank, Degree, and Number of Generators}}, booktitle = {38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)}, pages = {7:1--7:18}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.7}, URN = {urn:nbn:de:0030-drops-99068}, doi = {10.4230/LIPIcs.FSTTCS.2018.7}, annote = {Keywords: Combinatorial Nullstellensatz, Ideal Membership, Parametric Hardness, Low Rank Permanent} }

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

We describe a fixed parameter tractable (fpt) algorithm for Colored Hypergraph Isomorphism} which has running time $2^{O(b)}N^{O(1)}$, where the parameter $b$ is the maximum size of the color classes of the given hypergraphs and $N$ is the input size. We also describe fpt algorithms for certain permutation group problems that are used as subroutines in our algorithm.

V. Arvind, Bireswar Das, Johannes Köbler, and Seinosuke Toda. Colored Hypergraph Isomorphism is Fixed Parameter Tractable. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 327-337, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{arvind_et_al:LIPIcs.FSTTCS.2010.327, author = {Arvind, V. and Das, Bireswar and K\"{o}bler, Johannes and Toda, Seinosuke}, title = {{Colored Hypergraph Isomorphism is Fixed Parameter Tractable}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {327--337}, 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.2010.327}, URN = {urn:nbn:de:0030-drops-28751}, doi = {10.4230/LIPIcs.FSTTCS.2010.327}, annote = {Keywords: Fixed parameter tractability, fpt algorithms, graph isomorphism, computational complexity} }

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**Published in:** LIPIcs, Volume 2, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2008)

We study the algorithmic complexity of lattice problems based on the
sieving technique due to Ajtai, Kumar, and Sivakumar~\cite{aks}.
Given a $k$-dimensional subspace $M\subseteq \R^n$ and a full rank
integer lattice $\L\subseteq \Q^n$, the \emph{subspace avoiding
problem} SAP, defined by Bl\"omer and Naewe \cite{blomer}, is to
find a shortest vector in $\L\setminus M$. We first give a $2^{O(n+k
\log k)}$ time algorithm to solve \emph{the subspace avoiding
problem}. Applying this algorithm we obtain the following
results.
\begin{enumerate}
\item We give a $2^{O(n)}$ time algorithm to compute $i^{th}$
successive minima of a full rank lattice $\L\subset \Q^n$ if $i$ is
$O(\frac{n}{\log n})$.
\item We give a $2^{O(n)}$ time algorithm to solve a restricted
\emph{closest vector problem CVP} where the inputs fulfil a promise
about the distance of the input vector from the lattice.
\item We also show that unrestricted CVP has a $2^{O(n)}$ exact
algorithm if there is a $2^{O(n)}$ time exact algorithm for solving
CVP with additional input $v_i\in \L, 1\leq i\leq n$, where
$\|v_i\|_p$ is the $i^{th}$ successive minima of $\L$ for each $i$.
\end{enumerate}
We also give a new approximation algorithm for SAP and the
\emph{Convex Body Avoiding problem} which is a generalization of SAP.
Several of our algorithms work for \emph{gauge} functions as metric,
where the gauge function has a natural restriction and is accessed by
an oracle.

V. Arvind and Pushkar S. Joglekar. Some Sieving Algorithms for Lattice Problems. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 2, pp. 25-36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{arvind_et_al:LIPIcs.FSTTCS.2008.1738, author = {Arvind, V. and Joglekar, Pushkar S.}, title = {{Some Sieving Algorithms for Lattice Problems}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {25--36}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-08-8}, ISSN = {1868-8969}, year = {2008}, volume = {2}, editor = {Hariharan, Ramesh and Mukund, Madhavan and Vinay, V}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2008.1738}, URN = {urn:nbn:de:0030-drops-17380}, doi = {10.4230/LIPIcs.FSTTCS.2008.1738}, annote = {Keywords: Lattice problems, sieving algorithm, closest vector problem} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

We study the noncommutative rank problem, ncRANK, of computing the rank of matrices with linear entries in n noncommuting variables and the problem of noncommutative Rational Identity Testing, RIT, which is to decide if a given rational formula in n noncommuting variables is zero on its domain of definition.
Motivated by the question whether these problems have deterministic NC algorithms, we revisit their interrelationship from a parallel complexity point of view. We show the following results:
1) Based on Cohn’s embedding theorem [Cohn, 1990; Cohn, 2006] we show deterministic NC reductions from multivariate ncRANK to bivariate ncRANK and from multivariate RIT to bivariate RIT.
2) We obtain a deterministic NC-Turing reduction from bivariate RIT to bivariate ncRANK, thereby proving that a deterministic NC algorithm for bivariate ncRANK would imply that both multivariate RIT and multivariate ncRANK are in deterministic NC.

Vikraman Arvind and Pushkar S. Joglekar. A Multivariate to Bivariate Reduction for Noncommutative Rank and Related Results. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 14:1-14:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{arvind_et_al:LIPIcs.ICALP.2024.14, author = {Arvind, Vikraman and Joglekar, Pushkar S.}, title = {{A Multivariate to Bivariate Reduction for Noncommutative Rank and Related Results}}, booktitle = {51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)}, pages = {14:1--14:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-322-5}, ISSN = {1868-8969}, year = {2024}, volume = {297}, editor = {Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.14}, URN = {urn:nbn:de:0030-drops-201571}, doi = {10.4230/LIPIcs.ICALP.2024.14}, annote = {Keywords: noncommutative rank, rational formulas, identity testing, parallel complexity} }

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**Published in:** LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Based on a theorem of Bergman [Cohn, 2006] we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely, we show the following:
1) In the white-box setting, given an n-variate noncommutative polynomial f ∈ 𝔽⟨X⟩ over a field 𝔽 (either a finite field or the rationals) as an arithmetic circuit (or algebraic branching program), computing a complete factorization of f into irreducible factors is deterministic polynomial-time reducible to white-box factorization of a noncommutative bivariate polynomial g ∈ 𝔽⟨x,y⟩; the reduction transforms f into a circuit for g (resp. ABP for g), and given a complete factorization of g (namely, arithmetic circuits (resp. ABPs) for irreducible factors of g) the reduction recovers a complete factorization of f in polynomial time.
We also obtain a similar deterministic polynomial-time reduction in the black-box setting.
2) Additionally, we show over the field of rationals that bivariate linear matrix factorization of 4× 4 matrices is at least as hard as factoring square-free integers. This indicates that reducing noncommutative polynomial factorization to linear matrix factorization (as done in [Vikraman Arvind and Pushkar S. Joglekar, 2022]) is unlikely to succeed over the field of rationals even in the bivariate case. In contrast, multivariate linear matrix factorization for 3×3 matrices over rationals is in polynomial time.

Vikraman Arvind and Pushkar S. Joglekar. Multivariate to Bivariate Reduction for Noncommutative Polynomial Factorization. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{arvind_et_al:LIPIcs.MFCS.2023.14, author = {Arvind, Vikraman and Joglekar, Pushkar S.}, title = {{Multivariate to Bivariate Reduction for Noncommutative Polynomial Factorization}}, booktitle = {48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)}, pages = {14:1--14:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-292-1}, ISSN = {1868-8969}, year = {2023}, volume = {272}, editor = {Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.14}, URN = {urn:nbn:de:0030-drops-185480}, doi = {10.4230/LIPIcs.MFCS.2023.14}, annote = {Keywords: Arithmetic circuits, algebraic branching programs, polynomial factorization, automata, noncommutative polynomial ring} }

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**Published in:** LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring 𝔽∠{x_1,x_2,…,x_n} of polynomials in noncommuting variables x_1,x_2,…,x_n over the field 𝔽. We obtain the following result:
- We give a randomized algorithm that takes as input a noncommutative arithmetic formula of size s computing a noncommutative polynomial f ∈ 𝔽∠{x_1,x_2,…,x_n}, where 𝔽 = 𝔽_q is a finite field, and in time polynomial in s, n and log₂q computes a factorization of f as a product f = f_1f_2 ⋯ f_r, where each f_i is an irreducible polynomial that is output as a noncommutative algebraic branching program.
- The algorithm works by first transforming f into a linear matrix L using Higman’s linearization of polynomials. We then factorize the linear matrix L and recover the factorization of f. We use basic elements from Cohn’s theory of free ideals rings combined with Ronyai’s randomized polynomial-time algorithm for computing invariant subspaces of a collection of matrices over finite fields.

Vikraman Arvind and Pushkar S. Joglekar. On Efficient Noncommutative Polynomial Factorization via Higman Linearization. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 12:1-12:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{arvind_et_al:LIPIcs.CCC.2022.12, author = {Arvind, Vikraman and Joglekar, Pushkar S.}, title = {{On Efficient Noncommutative Polynomial Factorization via Higman Linearization}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {12:1--12:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-241-9}, ISSN = {1868-8969}, year = {2022}, volume = {234}, editor = {Lovett, Shachar}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.12}, URN = {urn:nbn:de:0030-drops-165747}, doi = {10.4230/LIPIcs.CCC.2022.12}, annote = {Keywords: Noncommutative Polynomials, Arithmetic Circuits, Factorization, Identity testing} }

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**Published in:** LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of polynomials over 𝔽₂ each of which is a product of affine forms.
We focus on the case of k-CNF formulas (the k-Sub-Sat problem). Clearly, k-Sub-Sat is no easier than k-SAT, and might be harder. Indeed, via simple reductions we show that 2-Sub-Sat is NP-hard, and W[1]-hard when parameterized by the co-dimension of the subspace. We also prove that the optimization version Max-2-Sub-Sat is NP-hard to approximate better than the trivial 3/4 ratio even on satisfiable instances.
On the algorithmic front, we investigate fast exponential algorithms which give non-trivial savings over brute-force algorithms. We give a simple branching algorithm with running time (1.5)^r for 2-Sub-Sat, where r is the subspace dimension, as well as an O^*(1.4312)ⁿ time algorithm where n is the number of variables.
Turning to k-Sub-Sat for k ⩾ 3, while known algorithms for solving a system of degree k polynomial equations already imply a solution with running time ≈ 2^{r(1-1/2k)}, we explore a more combinatorial approach. Based on an analysis of critical variables (a key notion underlying the randomized k-SAT algorithm of Paturi, Pudlak, and Zane), we give an algorithm with running time ≈ {n choose {⩽t}} 2^{n-n/k} where n is the number of variables and t is the co-dimension of the subspace. This improves upon the running time of the polynomial equations approach for small co-dimension. Our combinatorial approach also achieves polynomial space in contrast to the algebraic approach that uses exponential space. We also give a PPZ-style algorithm for k-Sub-Sat with running time ≈ 2^{n-n/2k}. This algorithm is in fact oblivious to the structure of the subspace, and extends when the subspace-membership constraint is replaced by any constraint for which partial satisfying assignments can be efficiently completed to a full satisfying assignment. Finally, for systems of O(n) polynomial equations in n variables over 𝔽₂, we give a fast exponential algorithm when each polynomial has bounded degree irreducible factors (but can otherwise have large degree) using a degree reduction trick.

Vikraman Arvind and Venkatesan Guruswami. CNF Satisfiability in a Subspace and Related Problems. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{arvind_et_al:LIPIcs.IPEC.2021.5, author = {Arvind, Vikraman and Guruswami, Venkatesan}, title = {{CNF Satisfiability in a Subspace and Related Problems}}, booktitle = {16th International Symposium on Parameterized and Exact Computation (IPEC 2021)}, pages = {5:1--5:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-216-7}, ISSN = {1868-8969}, year = {2021}, volume = {214}, editor = {Golovach, Petr A. and Zehavi, Meirav}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.5}, URN = {urn:nbn:de:0030-drops-153886}, doi = {10.4230/LIPIcs.IPEC.2021.5}, annote = {Keywords: CNF Satisfiability, Exact exponential algorithms, Hardness results} }

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**Published in:** LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)

Lubiw showed that several variants of Graph Isomorphism are NP-complete, where the solutions are required to satisfy certain additional constraints [SICOMP 10, 1981]. One of these, called Isomorphism With Restrictions, is to decide for two given graphs X_1=(V,E_1) and X_2=(V,E_2) and a subset R\subseteq V\times V of forbidden pairs whether there is an isomorphism \pi from X_1 to X_2 such that i^\pi\ne j for all (i,j)\in R. We prove that this problem and several of its generalizations are in fact in \FPT:
- The problem of deciding whether there is an isomorphism between two graphs that moves k vertices and satisfies Lubiw-style constraints is in FPT, with k and the size of R as parameters. The problem remains in FPT even if a conjunction of disjunctions of such constraints is allowed. As a consequence of the main result it follows that the problem to decide whether there is an isomorphism that moves exactly k vertices is in FPT. This solves a question left open in our article on exact weight automorphisms [STACS 2017].
- When the number of moved vertices is unrestricted, finding isomorphisms that satisfy a CNF of Lubiw-style constraints can be solved in FPT with access to a GI oracle.
- Checking if there is an isomorphism π between two graphs with complexity t is also in FPT with t as parameter, where the complexity of a permutation is the Cayley measure defined as the minimum number t such that \pi can be expressed as a product of t transpositions.
- We consider a more general problem in which the vertex set of a graph X is partitioned into Red and Blue, and we are interested in an automorphism that stabilizes Red and Blue and moves exactly k vertices in Blue, where k is the parameter. This problem was introduced by [Downey and Fellows 1999], and we showed [STACS 2017] that it is W[1]-hard even with color classes of size 4 inside Red. Now, for color classes of size at most 3 inside Red, we show the problem is in FPT.
In the non-parameterized setting, all these problems are NP-complete. Also, they all generalize in several ways the problem to decide whether there is an isomorphism between two graphs that moves at most k vertices, shown to be in FPT by Schweitzer [ESA 2011].

Vikraman Arvind, Johannes Köbler, Sebastian Kuhnert, and Jacobo Torán. Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 2:1-2:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{arvind_et_al:LIPIcs.IPEC.2017.2, author = {Arvind, Vikraman and K\"{o}bler, Johannes and Kuhnert, Sebastian and Tor\'{a}n, Jacobo}, title = {{Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {2:1--2:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-051-4}, ISSN = {1868-8969}, year = {2018}, volume = {89}, editor = {Lokshtanov, Daniel and Nishimura, Naomi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.2}, URN = {urn:nbn:de:0030-drops-85690}, doi = {10.4230/LIPIcs.IPEC.2017.2}, annote = {Keywords: parameterized algorithms, hypergraph isomorphism, mislabeled graphs} }

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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

In this paper we study arithmetic computations in the nonassociative, and noncommutative free polynomial ring F{X}. Prior to this work, nonassociative arithmetic computation was considered by Hrubes, Wigderson, and Yehudayoff, and they showed lower bounds and proved completeness results. We consider Polynomial Identity Testing and Polynomial Factorization in F{X} and show the following results.
1. Given an arithmetic circuit C computing a polynomial f in F{X} of degree d, we give a deterministic polynomial algorithm to decide if f is identically zero. Our result is obtained by a suitable adaptation of the PIT algorithm of Raz and Shpilka for noncommutative ABPs.
2. Given an arithmetic circuit C computing a polynomial f in F{X} of degree d, we give an efficient deterministic algorithm to compute circuits for the irreducible factors of f in polynomial time when F is the field of rationals. Over finite fields of characteristic p,
our algorithm runs in time polynomial in input size and p.

Vikraman Arvind, Rajit Datta, Partha Mukhopadhyay, and S. Raja. Efficient Identity Testing and Polynomial Factorization in Nonassociative Free Rings. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 38:1-38:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{arvind_et_al:LIPIcs.MFCS.2017.38, author = {Arvind, Vikraman and Datta, Rajit and Mukhopadhyay, Partha and Raja, S.}, title = {{Efficient Identity Testing and Polynomial Factorization in Nonassociative Free Rings}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {38:1--38:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.38}, URN = {urn:nbn:de:0030-drops-80690}, doi = {10.4230/LIPIcs.MFCS.2017.38}, annote = {Keywords: Circuits, Nonassociative, Noncommutative, Polynomial Identity Testing, Factorization} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

We show that checking if a given hypergraph has an automorphism that moves exactly k vertices is fixed parameter tractable, using k and additionally either the maximum hyperedge size or the maximum color class size as parameters. In particular, it suffices to use k as parameter if the hyperedge size is at most polylogarithmic in the size of the given hypergraph.
As a building block for our algorithms, we generalize Schweitzer's FPT algorithm [ESA 2011] that, given two graphs on the same vertex set and a parameter k, decides whether there is an isomorphism between the two graphs that moves at most k vertices. We extend this result to hypergraphs, using the maximum hyperedge size as a second parameter.
Another key component of our algorithm is an orbit-shrinking technique that preserves permutations that move few points and that may be of independent interest. Applying it to a suitable subgroup of the automorphism group allows us to switch from bounded hyperedge size to bounded color classes in the exactly-k case.

Vikraman Arvind, Johannes Köbler, Sebastian Kuhnert, and Jacobo Torán. Parameterized Complexity of Small Weight Automorphisms. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{arvind_et_al:LIPIcs.STACS.2017.7, author = {Arvind, Vikraman and K\"{o}bler, Johannes and Kuhnert, Sebastian and Tor\'{a}n, Jacobo}, title = {{Parameterized Complexity of Small Weight Automorphisms}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {7:1--7:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.7}, URN = {urn:nbn:de:0030-drops-70278}, doi = {10.4230/LIPIcs.STACS.2017.7}, annote = {Keywords: Parameterized algorithms, hypergraph isomorphism.} }

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

In this paper we study the complexity of the following problems:
1. Given a colored graph X=(V,E,c), compute a minimum cardinality set of vertices S (subset of V) such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G <= S_n given by generators, i.e., a minimum cardinality subset S of [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these problems. We show that when k=|S| is treated as parameter, then both problems are MINI[1]-hard. For the dual problems, where k=n-|S| is the parameter, we give FPT~algorithms.
2. A notion closely related to fixing is called individualization. Individualization combined with the Weisfeiler-Leman procedure is a fundamental technique in algorithms for Graph Isomorphism. Motivated by the power of individualization, in the present paper we explore the complexity of individualization: what is the minimum number of vertices we need to individualize in a given graph such that color refinement "succeeds" on it. Here "succeeds" could have different interpretations, and we consider the following: It could mean the individualized graph becomes: (a) discrete, (b) amenable, (c)compact, or (d) refinable. In particular, we study the parameterized versions of these problems where the parameter is the number of vertices individualized. We show a dichotomy: For graphs with color classes of size at most 3 these problems can be solved in polynomial time, while starting from color class size 4 they become W[P]-hard.

Vikraman Arvind, Frank Fuhlbrück, Johannes Köbler, Sebastian Kuhnert, and Gaurav Rattan. The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{arvind_et_al:LIPIcs.MFCS.2016.13, author = {Arvind, Vikraman and Fuhlbr\"{u}ck, Frank and K\"{o}bler, Johannes and Kuhnert, Sebastian and Rattan, Gaurav}, title = {{The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {13:1--13:14}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.13}, URN = {urn:nbn:de:0030-drops-64294}, doi = {10.4230/LIPIcs.MFCS.2016.13}, annote = {Keywords: parameterized complexity, graph automorphism, fixing number, base size, individualization} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

Using $\varepsilon$-bias spaces over $\F_2$, we show that the Remote Point Problem (RPP), introduced by Alon et al \cite{APY09}, has an $\NC^2$ algorithm (achieving the same parameters as \cite{APY09}). We study a generalization of the Remote Point Problem to groups: we replace $\F_2^n$ by $\mcG^n$ for an arbitrary fixed group $\mcG$. When $\mcG$ is Abelian we give an $\NC^2$ algorithm for RPP, again using $\varepsilon$-bias spaces. For nonabelian $\mcG$, we give a deterministic polynomial-time algorithm for RPP. We also show the connection to construction of expanding generator sets for the group $\mcG^n$. All our algorithms for the RPP achieve essentially the same parameters as \cite{APY09}.

Vikraman Arvind and Srikanth Srinivasan. The Remote Point Problem, Small Bias Spaces, and Expanding Generator Sets. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 59-70, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{arvind_et_al:LIPIcs.STACS.2010.2444, author = {Arvind, Vikraman and Srinivasan, Srikanth}, title = {{The Remote Point Problem, Small Bias Spaces, and Expanding Generator Sets}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {59--70}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2444}, URN = {urn:nbn:de:0030-drops-24449}, doi = {10.4230/LIPIcs.STACS.2010.2444}, annote = {Keywords: Small Bias Spaces, Expander Graphs, Cayley Graphs, Remote Point Problem} }

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**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

Motivated by the Hadamard product of matrices we define the Hadamard
product of multivariate polynomials and study its arithmetic circuit
and branching program complexity. We also give applications and
connections to polynomial identity testing. Our main results are
the following.
\begin{itemize}
\item[$\bullet$] We show that noncommutative polynomial identity testing for
algebraic branching programs over rationals is complete for
the logspace counting class $\ceql$, and over fields of characteristic
$p$ the problem is in $\ModpL/\Poly$.
\item[$\bullet$] We show an exponential lower bound for expressing the
Raz-Yehudayoff polynomial as the Hadamard product of two monotone
multilinear polynomials. In contrast the Permanent can be expressed
as the Hadamard product of two monotone multilinear formulas of
quadratic size.
\end{itemize}

Vikraman Arvind, Pushkar S. Joglekar, and Srikanth Srinivasan. Arithmetic Circuits and the Hadamard Product of Polynomials. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 25-36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{arvind_et_al:LIPIcs.FSTTCS.2009.2304, author = {Arvind, Vikraman and Joglekar, Pushkar S. and Srinivasan, Srikanth}, title = {{Arithmetic Circuits and the Hadamard Product of Polynomials}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {25--36}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2304}, URN = {urn:nbn:de:0030-drops-23046}, doi = {10.4230/LIPIcs.FSTTCS.2009.2304}, annote = {Keywords: Hadamard product, identity testing, lower bounds, algebraic branching programs} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

Motivated by the quantum algorithm for testing commutativity of black-box groups (Magniez and Nayak, 2007), we study the following problem: Given a black-box finite ring by an additive generating set and a multilinear polynomial over that ring, also accessed as a black-box function (we allow the indeterminates of the polynomial to be commuting or noncommuting), we study the problem of testing if the polynomial is an \emph{identity} for the given ring. We give a quantum algorithm with query complexity sub-linear in the number of generators for the ring, when the number of indeterminates of the input polynomial is small (ideally a constant). Towards a lower bound, we also show a reduction from a version of the collision problem (which is well studied in quantum computation) to a variant of this problem.

Vikraman Arvind and Partha Mukhopadhyay. Quantum Query Complexity of Multilinear Identity Testing. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 87-98, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{arvind_et_al:LIPIcs.STACS.2009.1801, author = {Arvind, Vikraman and Mukhopadhyay, Partha}, title = {{Quantum Query Complexity of Multilinear Identity Testing}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {87--98}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1801}, URN = {urn:nbn:de:0030-drops-18014}, doi = {10.4230/LIPIcs.STACS.2009.1801}, annote = {Keywords: Quantum algorithm, Identity testing, Query complexity, Multilinear polynomials} }