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Vikraman Arvind

Publications and source records attributed to Vikraman Arvind.

9 recordsLinked to original sources
arXiv2024

A Multivariate to Bivariate Reduction for Noncommutative Rank and Related Results

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 \cite{Co90,Cohnfir} 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.

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arXiv2022

Testing isomorphism of chordal graphs of bounded leafage is fixed-parameter tractable

The computational complexity of the graph isomorphism problem is considered to be a major open problem in theoretical computer science. It is known that testing isomorphism of chordal graphs is polynomial-time equivalent to the general graph isomorphism problem. Every chordal graph can be represented as the intersection graph of some subtrees of a representing tree, and the leafage of a chordal graph is defined to be the minimum number of leaves in a representing tree for it. We prove that chordal graph isomorphism is fixed parameter tractable with leafage as parameter. In the process we introduce the problem of isomorphism testing for higher-order hypergraphs and show that finding the automorphism group of order-$k$ hypergraphs with vertex color classes of size $b$ is fixed parameter tractable for any constant $k$ and $b$ as fixed parameter.

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arXiv2021

CNF Satisfiability in a Subspace and Related Problems

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 $\F_2$ each of which is a product of affine forms. We focus on the case of k-CNF formulas (the k-SUB-SAT problem). Clearly, it is no easier than k-SAT, and might be harder. Indeed, via simple reductions we show NP-hardness for k=2 and W[1]-hardness 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 runtime 1.5^r for 2-SUB-SAT, where $r$ is the subspace dimension and an O^*(1.4312)^n time algorithm where $n$ is the number of variables. For k more than 2, while known algorithms for solving a system of degree $k$ polynomial equations already imply a solution with runtime 2^{r(1-1/2k)}, we explore a more combinatorial approach. For instance, based on the notion of critical variables, we give an algorithm with running time ${n\choose {\le t}} 2^{n-n/k}$, 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 algorithm also achieves polynomial space in contrast to the algebraic approach that uses exponential space.

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arXiv2016

Identity Testing for +-Regular Noncommutative Arithmetic Circuits

An efficient randomized polynomial identity test for noncommutative polynomials given by noncommutative arithmetic circuits remains an open problem. The main bottleneck to applying known techniques is that a noncommutative circuit of size $s$ can compute a polynomial of degree exponential in $s$ with a double-exponential number of nonzero monomials. In this paper, we report some progress by dealing with two natural subcases (both allow for polynomials of exponential degree and a double exponential number of monomials): (1) We consider \emph{$+$-regular} noncommutative circuits: these are homogeneous noncommutative circuits with the additional property that all the $+$-gates are layered, and in each $+$-layer all gates have the same syntactic degree. We give a \emph{white-box} polynomial-time deterministic polynomial identity test for such circuits. Our algorithm combines some new structural results for $+$-regular circuits with known results for noncommutative ABP identity testing [RS05PIT], rank bound of commutative depth three identities [SS13], and equivalence testing problem for words [Loh15, MSU97, Pla94]. (2) Next, we consider $ΣΠ^*Σ$ noncommutative circuits: these are noncommutative circuits with layered $+$-gates such that there are only two layers of $+$-gates. These $+$-layers are the output $+$-gate and linear forms at the bottom layer; between the $+$-layers the circuit could have any number of $\times$ gates. We given an efficient randomized \emph{black-box} identity testing problem for $ΣΠ^*Σ$ circuits. In particular, we show if $f\in F $ is a nonzero noncommutative polynomial computed by a $ΣΠ^*Σ$ circuit of size $s$, then $f$ cannot be a polynomial identity for the matrix algebra $\mathbb{M}_s(F)$, where the field $F$ is a sufficiently large extension of $F$ depending on the degree of $f$.

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arXiv2013

The Parameterized Complexity of some Permutation Group Problems

In this paper we study the parameterized complexity of two well-known permutation group problems which are NP-complete. 1. Given a permutation group G= , subgroup of $S_n$, and a parameter $k$, find a permutation $π$ in G such that $|{i\in [n]\mid π(i)\ne i}|$ is at least $k$. This generalizes the well-known NP-complete problem of finding a fixed-point free permutation in G. (this is the case when $k=n$). We show that this problem with parameter $k$ is fixed parameter tractable. In the process, we give a simple deterministic polynomial-time algorithm for finding a fixed point free element in a transitive permutation group, answering an open question of Cameron. 2. Next we consider the problem of computing a base for a permutation group G= . A base for G is a subset B of $[n]$ such that the subgroup of G that fixes B pointwise is trivial. This problem is known to be NP-complete. We show that it is fixed parameter tractable for the case of cyclic permutation groups and for permutation groups of constant orbit size. For more general classes of permutation groups we do not know whether the problem is in FPT or is W[1]-hard.

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arXiv2010

The Remote Point Problem, Small Bias Space, and Expanding Generator Sets

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

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arXiv2009

Circuit Lower Bounds, Help Functions, and the Remote Point Problem

We investigate the power of Algebraic Branching Programs (ABPs) augmented with help polynomials, and constant-depth Boolean circuits augmented with help functions. We relate the problem of proving explicit lower bounds in both these models to the Remote Point Problem (introduced by Alon, Panigrahy, and Yekhanin (RANDOM '09)). More precisely, proving lower bounds for ABPs with help polynomials is related to the Remote Point Problem w.r.t. the rank metric, and for constant-depth circuits with help functions it is related to the Remote Point Problem w.r.t. the Hamming metric. For algebraic branching programs with help polynomials with some degree restrictions we show exponential size lower bounds for explicit polynomials.

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