Eric Allender, Robert Beals, Mitsunori Ogihara

We characterize the complexity of some natural and important

problems in linear algebra. In particular, we identify natural

complexity classes for which the problems of (a) determining if a

system of linear equations is feasible and (b) computing the rank of

an integer matrix, ...
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Meena Mahajan, V Vinay

We prove a new combinatorial characterization of the

determinant. The characterization yields a simple

combinatorial algorithm for computing the

determinant. Hitherto, all (known) algorithms for

determinant have been based on linear algebra. Our

combinatorial algorithm requires no division and works over

arbitrary commutative rings. It also lends itself to

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Meena Mahajan, V Vinay

In this paper we approach the problem of computing the characteristic

polynomial of a matrix from the combinatorial viewpoint. We present

several combinatorial characterizations of the coefficients of the

characteristic polynomial, in terms of walks and closed walks of

different kinds in the underlying graph. We develop algorithms based

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Eric Allender, Klaus Reinhardt

We show that the perfect matching problem is in the

complexity class SPL (in the nonuniform setting).

This provides a better upper bound on the complexity of the

matching problem, as well as providing motivation for studying

the complexity class SPL.

Using similar ...
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Eric Allender, Shiyu Zhou

We show that the complexity class LogFew is contained

in NL $\cap$ SPL. Previously, this was known only to

hold in the nonuniform setting.

Eric Allender, Vikraman Arvind, Meena Mahajan

The aim of this paper is to use formal power series techniques to

study the structure of small arithmetic complexity classes such as

GapNC^1 and GapL. More precisely, we apply the Kleene closure of

languages and the formal power series operations of inversion and

root ...
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Meena Mahajan, P R Subramanya, V Vinay

The Pfaffian of an oriented graph is closely linked to

Perfect Matching. It is also naturally related to the determinant of

an appropriately defined matrix. This relation between Pfaffian and

determinant is usually exploited to give a fast algorithm for

computing Pfaffians.

We present the first completely combinatorial algorithm for ... more >>>

Meena Mahajan, V Vinay

In this note, we consider the problem of computing the

coefficients of the characteristic polynomial of a given

matrix, and the related problem of verifying the

coefficents.

Santha and Tan [CC98] show that verifying the determinant

(the constant term in the characteristic polynomial) is

complete for the class C=L, ...
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Alina Beygelzimer, Mitsunori Ogihara

We investigate the complexity of enumerative approximation of

two elementary problems in linear algebra, computing the rank

and the determinant of a matrix. In particular, we show that

if there exists an enumerator that, given a matrix, outputs a

list of constantly many numbers, one of which is guaranteed to

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Nitin Saxena

In this paper we give the first deterministic polynomial time algorithm for testing whether a {\em diagonal} depth-$3$ circuit $C(\arg{x}{n})$ (i.e. $C$ is a sum of powers of linear functions) is zero. We also prove an exponential lower bound showing that such a circuit will compute determinant or permanent only ... more >>>

Vikraman Arvind, Srikanth Srinivasan

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:

... more >>>Neeraj Kayal

An $m$-variate polynomial $f$ is said to be an affine projection of some $n$-variate polynomial $g$ if there exists an $n \times m$ matrix $A$ and an $n$-dimensional vector $b$ such that $f(x) = g(A x + b)$. In other words, if $f$ can be obtained by replacing each variable ... more >>>

Joshua Grochow

We study the problem of matrix Lie algebra conjugacy. Lie algebras arise centrally in areas as diverse as differential equations, particle physics, group theory, and the Mulmuley--Sohoni Geometric Complexity Theory program. A matrix Lie algebra is a set $\mathcal{L}$ of matrices such that $A,B \in \mathcal{L}$ implies$AB - BA \in ... more >>>

Pavel Hrubes, Iddo Tzameret

We study arithmetic proof systems $\mathbb{P}_c(\mathbb{F})$ and $\mathbb{P}_f(\mathbb{F})$ operating with arithmetic circuits and arithmetic formulas, respectively, that prove polynomial identities over a field $\mathbb{F}$. We establish a series of structural theorems about these proof systems, the main one stating that $\mathbb{P}_c(\mathbb{F})$ proofs can be balanced: if a polynomial identity of ... more >>>

Ankit Gupta, Pritish Kamath, Neeraj Kayal, Ramprasad Saptharishi

Agrawal and Vinay (FOCS 2008) have recently shown that an exponential lower bound for depth four homogeneous circuits with bottom layer of $\times$ gates having sublinear fanin translates to an exponential lower bound for a general arithmetic circuit computing the permanent. Motivated by this, we examine the complexity of computing ... more >>>

Markus Bläser

We consider the complexity of computing the determinant over arbitrary finite-dimensional algebras. We first consider the case that $A$ is fixed. We obtain the following dichotomy: If $A/rad(A)$ is noncommutative, then computing the determinant over $A$ is hard. ``Hard'' here means $\#P$-hard over fields of characteristic $0$ and $ModP_p$-hard over ... more >>>

Ankit Gupta, Pritish Kamath, Neeraj Kayal, Ramprasad Saptharishi

We show that, over $\mathbb{C}$, if an $n$-variate polynomial of degree $d = n^{O(1)}$ is computable by an arithmetic circuit of size $s$ (respectively by an algebraic branching program of size $s$) then it can also be computed by a depth three circuit (i.e. a $\Sigma \Pi \Sigma$-circuit) of size ... more >>>

Craig Gentry

We show that, for many noncommutative algebras A, the hardness of computing the determinant of matrices over A follows almost immediately from Barrington’s Theorem. Barrington showed how to express a NC1 circuit as a product program over any non-solvable group. We construct a simple matrix directly from Barrington’s product program ... more >>>

Pushkar Joglekar, Aravind N.R.

We introduce and study the notion of read-$k$ projections of the determinant: a polynomial $f \in \mathbb{F}[x_1, \ldots, x_n]$ is called a {\it read-$k$ projection of determinant} if $f=det(M)$, where entries of matrix $M$ are either field elements or variables such that each variable appears at most $k$ times in ... more >>>