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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} }

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