arXiv subjects

V. Arvind

Publications and source records attributed to V. Arvind.

45 records · Page 3Linked to original sources

arXiv2008

Quantum Query Complexity of Multilinear Identity Testing

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 $O(m(1+α)^{m/2} k^{\frac{m}{m+1}})$ assuming $k\geq (1+1/α)^{m+1}$. 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.

cs.CC↗

arXiv2008

Lattice Problems, Gauge Functions and Parameterized Algorithms

Given a k-dimensional subspace M\subseteq \R^n and a full rank integer lattice L\subseteq \R^n, the \emph{subspace avoiding problem} SAP is to find a shortest vector in L\setminus M. Treating k as a parameter, we obtain new parameterized approximation and exact algorithms for SAP based on the AKS sieving technique. More precisely, we give a randomized $(1+ε)$-approximation algorithm for parameterized SAP that runs in time 2^{O(n)}.(1/ε)^k, where the parameter k is the dimension of the subspace M. Thus, we obtain a 2^{O(n)} time algorithm for ε=2^{-O(n/k)}. We also give a 2^{O(n+k\log k)} exact algorithm for the parameterized SAP for any \ell_p norm. Several of our algorithms work for all gauge functions as metric with some natural restrictions, in particular for all \ell_p norms. We also prove an Ω(2^n) lower bound on the query complexity of AKS sieving based exact algorithms for SVP that accesses the gauge function as oracle.

cs.CC↗

arXiv2008

Derandomizing the Isolation Lemma and Lower Bounds for Circuit Size

The isolation lemma of Mulmuley et al \cite{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 \emph{deterministic} identity tests has received a lot of attention recently. The goal of this note 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. The restricted versions of the isolation lemma we consider are natural and would suffice for the standard applications of the isolation lemma. 2. From the result of Klivans-Spielman \cite{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 also implies circuit size lower bounds in the commutative setting.

cs.CC↗

arXiv2008

New results on Noncommutative and Commutative Polynomial Identity Testing

Using ideas from automata theory we design a new efficient (deterministic) identity test for the \emph{noncommutative} polynomial identity testing problem (first introduced and studied in \cite{RS05,BW05}). We also apply this idea to the reconstruction of black-box noncommuting algebraic branching programs. Assuming the black-box model allows us to query the ABP for the output at any given gate, we can reconstruct an (equivalent) ABP in deterministic polynomial time. Finally, we explore commutative identity testing when the coefficients of the input polynomial come from an arbitrary finite commutative ring with unity.

cs.CC↗

arXiv2007

On Computing the Distinguishing Numbers of Planar Graphs and Beyond: a Counting Approach

A vertex k-labeling of graph G is distinguishing if the only automorphism that preserves the labels of G is the identity map. The distinguishing number of G, D(G), is the smallest integer k for which G has a distinguishing k-labeling. In this paper, we apply the principle of inclusion-exclusion and develop recursive formulas to count the number of inequivalent distinguishing k-labelings of a graph. Along the way, we prove that the distinguishing number of a planar graph can be computed in time polynomial in the size of the graph.}

math.CO↗

arXiv2006

A Polynomial Time Nilpotence Test for Galois Groups and Related Results

We give a deterministic polynomial-time algorithm to check whether the Galois group $\Gal{f}$ of an input polynomial $f(X) \in \Q[X]$ is nilpotent: the running time is polynomial in $\size{f}$. Also, we generalize the Landau-Miller solvability test to an algorithm that tests if $\Gal{f}$ is in $Γ_d$: this algorithm runs in time polynomial in $\size{f}$ and $n^d$ and, moreover, if $\Gal{f}\inΓ_d$ it computes all the prime factors of $# \Gal{f}$.

cs.CC↗

arXiv2002

The Quantum Query Complexity of 0-1 Knapsack and Associated Claw Problems

We first give an $Ø(2^{n/3})$ quantum algorithm for the 0-1 Knapsack problem with $n$ variables. More generally, for 0-1 Integer Linear Programs with $n$ variables and $d$ inequalities we give an $Ø(2^{n/3}n^d)$ quantum algorithm. For $d =o(n/\log n)$ this running time is bounded by $Ø(2^{n(1/3+ε)})$ for every $ε>0$ and in particular it is better than the $Ø(2^{n/2})$ upper bound for general quantum search. To investigate whether better algorithms for these NP-hard problems are possible, we formulate a \emph{symmetric} claw problem corresponding to 0-1 Knapsack and study its quantum query complexity. For the symmetric claw problem we establish a lower bound of $Ø(2^{n/4})$ for its quantum query complexity. We have an $Ø(2^{n/3})$ upper bound given by essentially the same quantum algorithm that works for Knapsack. Additionally, we consider CNF satisfiability of CNF formulas $F$ with no restrictions on clause size, but with the number of clauses in $F$ bounded by $cn$ for a constant $c$, where $n$ is the number of variables. We give a $2^{(1-α)n/2}$ quantum algorithm for satisfiability in this case, where $α$ is a constant depending on $c$.

quant-ph↗

arXiv2002

Nonstabilizer Quantum Codes from Abelian Subgroups of the Error Group

This paper is motivated by the computer-generated nonadditive ((5,6,2)) code described in an article by Rains, Hardin, Shor and Sloane. We describe a theory of non-stabilizer codes of which the nonadditive code of Rains et al is an example. Furthermore, we give a general strategy of constructing good nonstabilizer codes from good stabilizer codes and give some explicit constructions and asymptotically good nonstabilizer codes. In fact, we explicitly construct a family of distance 2 non-stabilizer codes over all finite fields of which the ((5,6,2)) is an special example. More interestingly, using our theory, we are also able to explicitly construct examples of non-stablizer quantum codes of distance 3. Like in the case of stabilizer codes, we can design fairly efficient encoding and decoding procedures.

quant-ph↗

arXiv2002

A Family of Quantum Stabilizer Codes Based on the Weyl Commutation Relations over a Finite Field

Using the Weyl commutation relations over a finite field we introduce a family of error-correcting quantum stabilizer codes based on a class of symmetric matrices over the finite field satisfying certain natural conditions. When the field is GF(2) the existence of a rich class of such symmetric matrices is demonstrated by a simple probabilistic argument depending on the Chernoff bound for i.i.d symmetric Bernoulli trials. If, in addition, these symmetric matrices are assumed to be circulant it is possible to obtain concrete examples by a computer program. The quantum codes thus obtained admit elegant encoding circuits.

quant-ph↗