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Electronic Colloquium on Computational Complexity

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All reports by Author Nutan Limaye:

TR18-062 | 7th April 2018
Suryajith Chillara, Christian Engels, Nutan Limaye, Srikanth Srinivasan

A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits

We study the size blow-up that is necessary to convert an algebraic circuit of product-depth $\Delta+1$ to one of product-depth $\Delta$ in the multilinear setting.

We show that for every positive $\Delta = \Delta(n) = o(\log n/\log \log n),$ there is an explicit multilinear polynomial $P^{(\Delta)}$ on $n$ variables that ... more >>>

TR17-156 | 15th October 2017
Suryajith Chillara, Nutan Limaye, Srikanth Srinivasan

Small-depth Multilinear Formula Lower Bounds for Iterated Matrix Multiplication, with Applications

The complexity of Iterated Matrix Multiplication is a central theme in Computational Complexity theory, as the problem is closely related to the problem of separating various complexity classes within $\mathrm{P}$. In this paper, we study the algebraic formula complexity of multiplying $d$ many $2\times 2$ matrices, denoted $\mathrm{IMM}_{d}$, and show ... more >>>

TR17-077 | 30th April 2017
Guillaume Lagarde, Nutan Limaye, Srikanth Srinivasan

Lower Bounds and PIT for Non-Commutative Arithmetic circuits with Restricted Parse Trees

We investigate the power of Non-commutative Arithmetic Circuits, which compute polynomials over the free non-commutative polynomial ring $\mathbb{F}\langle x_1,\dots,x_N \rangle$, where variables do not commute. We consider circuits that are restricted in the ways in which they can compute monomials: this can be seen as restricting the families of parse ... more >>>

TR17-019 | 8th February 2017
Andreas Krebs, Nutan Limaye, Michael Ludwig

A Unified Method for Placing Problems in Polylogarithmic Depth

Revisions: 2

In this work we consider the term evaluation problem which involves, given a term over some algebra and a valid input to the term, computing the value of the term on that input. This is a classical problem studied under many names such as formula evaluation problem, formula value problem ... more >>>

TR16-155 | 10th October 2016
Vaibhav Krishan, Nutan Limaye

Isolation Lemma for Directed Reachability and NL vs. L

In this work we study the problem of efficiently isolating witnesses for the complexity classes NL and LogCFL, which are two well-studied complexity classes contained in P. We prove that if there is a L/poly randomized procedure with success probability at least 2/3 for isolating an s-t path in a ... more >>>

TR16-143 | 15th September 2016
Nikhil Balaji, Nutan Limaye, Srikanth Srinivasan

An Almost Cubic Lower Bound for $\Sigma\Pi\Sigma$ Circuits Computing a Polynomial in VP

In this note, we prove that there is an explicit polynomial in VP such that any $\Sigma\Pi\Sigma$ arithmetic circuit computing it must have size at least $n^{3-o(1)}$. Up to $n^{o(1)}$ factors, this strengthens a recent result of Kayal, Saha and Tavenas (ICALP 2016) which gives a polynomial in VNP with ... more >>>

TR15-118 | 23rd July 2015
Hervé Fournier, Nutan Limaye, Meena Mahajan, Srikanth Srinivasan

The shifted partial derivative complexity of Elementary Symmetric Polynomials

We continue the study of the shifted partial derivative measure, introduced by Kayal (ECCC 2012), which has been used to prove many strong depth-4 circuit lower bounds starting from the work of Kayal, and that of Gupta et al. (CCC 2013).

We show a strong lower bound on the dimension ... more >>>

TR15-022 | 9th February 2015
Nutan Limaye, Guillaume Malod, Srikanth Srinivasan

Lower bounds for non-commutative skew circuits

Revisions: 1

Nisan (STOC 1991) exhibited a polynomial which is computable by linear sized non-commutative circuits but requires exponential sized non-commutative algebraic branching programs. Nisan's hard polynomial is in fact computable by linear sized skew circuits (skew circuits are circuits where every multiplication gate has the property that all but one of ... more >>>

TR14-183 | 25th December 2014
Nikhil Balaji, Andreas Krebs, Nutan Limaye

Skew Circuits of Small Width

A celebrated result of Barrington (1985) proved that polynomial size, width-5 branching programs (BP) are equivalent in power to a restricted form of branching programs -- polynomial sized width-5 permutation branching programs (PBP), which in turn capture all of NC1. On the other hand it is known that width-3 PBPs ... more >>>

TR14-180 | 22nd December 2014
Anna Gal, Jing-Tang Jang, Nutan Limaye, Meena Mahajan, Karteek Sreenivasaiah

Space-Efficient Approximations for Subset Sum

SUBSET SUM is a well known NP-complete problem:
given $t \in Z^{+}$ and a set $S$ of $m$ positive integers, output YES if and only if there is a subset $S^\prime \subseteq S$ such that the sum of all numbers in $S^\prime$ equals $t$. The problem and its search ... more >>>

TR14-005 | 14th January 2014
Neeraj Kayal, Nutan Limaye, Chandan Saha, Srikanth Srinivasan

An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

We show here a $2^{\Omega(\sqrt{d} \cdot \log N)}$ size lower bound for homogeneous depth four arithmetic formulas. That is, we give
an explicit family of polynomials of degree $d$ on $N$ variables (with $N = d^3$ in our case) with $0, 1$-coefficients such that
for any representation of ... more >>>

TR13-102 | 17th July 2013
Andreas Krebs, Nutan Limaye, Meena Mahajan, Karteek Sreenivasaiah

Small Depth Proof Systems

A proof system for a language $L$ is a function $f$ such that Range$(f)$ is exactly $L$. In this paper, we look at proofsystems from a circuit complexity point of view and study proof systems that are computationally very restricted. The restriction we study is: they can be computed by ... more >>>

TR13-100 | 15th July 2013
Hervé Fournier, Nutan Limaye, Guillaume Malod, Srikanth Srinivasan

Lower bounds for depth $4$ formulas computing iterated matrix multiplication

We study the arithmetic complexity of iterated matrix multiplication. We show that any multilinear homogeneous depth $4$ arithmetic formula computing the product of $d$ generic matrices of size $n \times n$, IMM$_{n,d}$, has size $n^{\Omega(\sqrt{d})}$ as long as $d \leq n^{1/10}$. This improves the result of Nisan and Wigderson (Computational ... more >>>

TR12-186 | 27th December 2012
Andreas Krebs, Nutan Limaye

DLOGTIME-Proof Systems

We define DLOGTIME proof systems, DLTPS, which generalize NC0 proof systems.
It is known that functions such as Exact-k and Majority do not have NC0 proof systems. Here, we give a DLTPS for Exact-k (and therefore for Majority) and also for other natural functions such as Reach and k-Clique. Though ... more >>>

TR10-103 | 28th June 2010
Andreas Krebs, Nutan Limaye, Meena Mahajan

Counting paths in VPA is complete for \#NC$^1$

We give a \#NC$^1$ upper bound for the problem of counting accepting paths in any fixed visibly pushdown automaton. Our algorithm involves a non-trivial adaptation of the arithmetic formula evaluation algorithm of Buss, Cook, Gupta, Ramachandran (BCGR: SICOMP 21(4), 1992). We also show that the problem is \#NC$^1$ hard. Our ... more >>>

TR10-094 | 17th May 2010
Ajesh Babu, Nutan Limaye, Girish Varma

Streaming algorithms for some problems in log-space.

In this paper, we give streaming algorithms for some problems which are known to be in deterministic log-space, when the number of passes made on the input is unbounded. If the input data is massive,
the conventional deterministic log-space algorithms may not run efficiently. We study the complexity of the ... more >>>

TR09-052 | 2nd May 2009
Fabian Wagner, Samir Datta, Nutan Limaye, Prajakta Nimbhorkar, Thomas Thierauf

Planar Graph Isomorphism is in Log-space

Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. There is a significant gap between extant lower and upper bounds for planar graphs as well. We bridge the gap for this natural and ... more >>>

TR07-087 | 11th July 2007
Nutan Limaye, Meena Mahajan, B. V. Raghavendra Rao

Arithmetizing classes around NC^1 and L

The parallel complexity class NC^1 has many equivalent models such as
polynomial size formulae and bounded width branching
programs. Caussinus et al. \cite{CMTV} considered arithmetizations of
two of these classes, #NC^1 and #BWBP. We further this study to
include arithmetization of other classes. In particular, we show that
counting paths ... more >>>

TR06-009 | 10th January 2006
Nutan Limaye, Meena Mahajan, Jayalal Sarma

Evaluating Monotone Circuits on Cylinders, Planes and Tori

We re-examine the complexity of evaluating monotone planar circuits
MPCVP, with special attention to circuits with cylindrical
embeddings. MPCVP is known to be in NC^3, and for the special
case of upward stratified circuits, it is known to be in
LogDCFL. We characterize cylindricality, which ... more >>>

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