Publications (9)0 Total impact
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Article: Near-Optimal Expanding Generating Sets for Solvable Permutation Groups
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ABSTRACT: Let $G = $ be a solvable permutation group of the symmetric group $S_n$ given as input by the generating set $S$. We give a deterministic polynomial-time algorithm that computes an \emph{expanding generating set} of size $\tilde{O}(n^2)$ for $G$. More precisely, the algorithm computes a subset $T\subset G$ of size $\tilde{O}(n^2)(1/\lambda)^{O(1)}$ such that the undirected Cayley graph $Cay(G,T)$ is a $\lambda$-spectral expander (the $\tilde{O}$ notation suppresses $\log ^{O(1)}n$ factors). As a byproduct of our proof, we get a new explicit construction of $\varepsilon$-bias spaces of size $\tilde{O}(n\poly(\log d))(\frac{1}{\varepsilon})^{O(1)}$ for the groups $\Z_d^n$. The earlier known size bound was $O((d+n/\varepsilon^2))^{11/2}$ given by \cite{AMN98}.01/2012; -
Article: On the hardness of the noncommutative determinant
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ABSTRACT: In this paper we study the computational complexity of computing the noncommutative determinant. We first consider the arithmetic circuit complexity of computing the noncommutative determinant polynomial. Then, more generally, we also examine the complexity of computing the determinant (as a function) over noncommutative domains. Our hardness results are summarized below: 1. We show that if the noncommutative determinant polynomial has small noncommutative arithmetic circuits then so does the noncommutative permanent. Consequently, the commutative permanent polynomial has small commutative arithmetic circuits. 2. For any field F we show that computing the n X n permanent over F is polynomial-time reducible to computing the 2n X 2n (noncommutative) determinant whose entries are O(n^2) X O(n^2) matrices over the field F. 3. We also derive as a consequence that computing the n X n permanent over nonnegative rationals is polynomial-time reducible to computing the noncommutative determinant over Clifford algebras of n^{O(1)} dimension. Our techniques are elementary and use primarily the notion of the Hadamard Product of noncommutative polynomials. Comment: 11 pages, v2: 18 pages, some typos removed, new section added on Clifford algebras, and some reorganization10/2009; -
Article: Arithmetic Circuits and the Hadamard Product of Polynomials
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ABSTRACT: 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. 1. 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 $\ModpL/\Poly$. 2.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. Comment: 20 pages07/2009; -
Article: On Lower Bounds for Constant Width Arithmetic Circuits
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ABSTRACT: The motivation for this paper is to study the complexity of constant-width arithmetic circuits. Our main results are the following. 1. For every k > 1, we provide an explicit polynomial that can be computed by a linear-sized monotone circuit of width 2k but has no subexponential-sized monotone circuit of width k. It follows, from the definition of the polynomial, that the constant-width and the constant-depth hierarchies of monotone arithmetic circuits are infinite, both in the commutative and the noncommutative settings. 2. We prove hardness-randomness tradeoffs for identity testing constant-width commutative circuits analogous to [KI03,DSY08]. Comment: 16 pages07/2009; -
Chapter: On Lower Bounds for Constant Width Arithmetic Circuits
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ABSTRACT: For every k > 1, we give an explicit polynomial that is computable by a linear-sized monotone circuit of width 2k but has no subexponential-sized monotone circuit of width k. As a consequence we show that the constant-width and the constant-depth hierarchies of monotone arithmetic circuits (both commutative and noncommutative) are infinite. We also prove hardness-randomness tradeoffs for identity testing of constant-width circuits analogous to [6,4].04/2009: pages 637-646; -
Chapter: Derandomizing the Isolation Lemma and Lower Bounds for Circuit Size
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ABSTRACT: The isolation lemma of Mulmuley et al [MVV87] is an important tool in the design of randomized algorithms and has played an important role in several nontrivial complexity upper bounds. On the other hand, polynomial identity testing is a well-studied algorithmic problem with efficient randomized algorithms and the problem of obtaining efficient deterministic identity tests has received a lot of attention recently. The goal of this paper is to compare the isolation lemma with polynomial identity testing: 1 We show that derandomizing reasonably restricted versions of the isolation lemma implies circuit size lower bounds. We derive the circuit lower bounds by examining the connection between the isolation lemma and polynomial identity testing. We give a randomized polynomial-time identity test for noncommutative circuits of polynomial degree based on the isolation lemma. Using this result, we show that derandomizing the isolation lemma implies noncommutative circuit size lower bounds. For the commutative case, a stronger derandomization hypothesis allows us to construct an explicit multilinear polynomial that does not have subexponential size commutative circuits. The restricted versions of the isolation lemma we consider are natural and would suffice for the standard applications of the isolation lemma. 1 From the result of Klivans-Spielman [KS01] we observe that there is a randomized polynomial-time identity test for commutative circuits of polynomial degree, also based on a more general isolation lemma for linear forms. Consequently, derandomization of (a suitable version of) this isolation lemma implies that either or the Permanent over does not have polynomial-size arithmetic circuits.08/2008: pages 276-289; -
Article: Quantum Query Complexity of Multilinear Identity Testing
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ABSTRACT: Motivated by the quantum algorithm in \cite{MN05} for testing commutativity of black-box groups, we study the following problem: Given a black-box finite ring $R=\angle{r_1,...,r_k}$ where $\{r_1,r_2,...,r_k\}$ is an additive generating set for $R$ and a multilinear polynomial $f(x_1,...,x_m)$ over $R$ also accessed as a black-box function $f:R^m\to R$ (where we allow the indeterminates $x_1,...,x_m$ to be commuting or noncommuting), we study the problem of testing if $f$ is an \emph{identity} for the ring $R$. More precisely, the problem is to test if $f(a_1,a_2,...,a_m)=0$ for all $a_i\in R$. We give a quantum algorithm with query complexity $O(m(1+\alpha)^{m/2} k^{\frac{m}{m+1}})$ assuming $k\geq (1+1/\alpha)^{m+1}$. Towards a lower bound, we also discuss a reduction from a version of $m$-collision to this problem. We also observe a randomized test with query complexity $4^mmk$ and constant success probability and a deterministic test with $k^m$ query complexity.08/2008; -
Chapter: On Isomorphism and Canonization of Tournaments and Hypertournaments
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ABSTRACT: We give a polynomial-time oracle algorithm for Tournament Canonization that accesses oracles for Tournament Isomorphism and Rigid-Tournament Canonization. Extending the Babai-Luks Tournament Canonization algorithm, we give an n O(k + logn) algorithm for canonization and isomorphism testing of k-hypertournaments, where n is the number of vertices and k is the size of hyperedges.11/2006: pages 449-459; -
Article: Some Sieving Algorithms for Lattice Problems
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ABSTRACT: 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 $Msubseteq R^n$ and a full rank integer lattice $Lsubseteq Q^n$, the emph{subspace avoiding problem} SAP, defined by Bl"omer and Naewe cite{blomer}, is to find a shortest vector in $Lsetminus M$. 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 $Lsubset Q^n$ 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 $v_iin L, 1leq ileq n$, 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. @InProceedings{arvind_et_al:LIPIcs:2008:1738, author = {V. Arvind and Pushkar S. Joglekar}, title = {Some Sieving Algorithms for Lattice Problems}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2008)}, series = {Leibniz International Proceedings in Informatics}, year = {2008}, volume = {2}, editor = {Ramesh Hariharan and Madhavan Mukund and V Vinay}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany}, address = {Dagstuhl, Germany}, URL = {http://drops.dagstuhl.de/opus/volltexte/2008/1738}, URN = {urn:nbn:de:0030-drops-17380}, annote = {Keywords: Lattice problems, sieving algorithm, closest vector problem} }
Top co-authors
Institutions
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2006–2009
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Chennai Mathematical Institute
Chennai, State of Tamil Nadu, India
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