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Electronic Colloquium on Computational Complexity

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All reports by Author Rahul Jain:

TR17-123 | 2nd August 2017
Dmitry Gavinsky, Rahul Jain, Hartmut Klauck, Srijita Kundu, Troy Lee, Miklos Santha, Swagato Sanyal, Jevgenijs Vihrovs

Quadratically Tight Relations for Randomized Query Complexity

Let $f:\{0,1\}^n \rightarrow \{0,1\}$ be a Boolean function. The certificate complexity $C(f)$ is a complexity measure that is quadratically tight for the zero-error randomized query complexity $R_0(f)$: $C(f) \leq R_0(f) \leq C(f)^2$. In this paper we study a new complexity measure that we call expectational certificate complexity $EC(f)$, which is ... more >>>

TR17-107 | 1st June 2017
Anurag Anshu, Dmitry Gavinsky, Rahul Jain, Srijita Kundu, Troy Lee, Priyanka Mukhopadhyay, Miklos Santha, Swagato Sanyal

A Composition Theorem for Randomized Query complexity

Revisions: 1

Let the randomized query complexity of a relation for error probability $\epsilon$ be denoted by $\R_\epsilon(\cdot)$. We prove that for any relation $f \subseteq \{0,1\}^n \times \mathcal{R}$ and Boolean function $g:\{0,1\}^m \rightarrow \{0,1\}$, $\R_{1/3}(f\circ g^n) = \Omega(\R_{4/9}(f)\cdot\R_{1/2-1/n^4}(g))$, where $f \circ g^n$ is the relation obtained by composing $f$ and $g$. ... more >>>

TR17-054 | 22nd March 2017
Anurag Anshu, Naresh Goud, Rahul Jain, Srijita Kundu, Priyanka Mukhopadhyay

Lifting randomized query complexity to randomized communication complexity

Revisions: 3

We show that for any (partial) query function $f:\{0,1\}^n\rightarrow \{0,1\}$, the randomized communication complexity of $f$ composed with $\mathrm{Index}^n_m$ (with $m= \poly(n)$) is at least the randomized query complexity of $f$ times $\log n$. Here $\mathrm{Index}_m : [m] \times \{0,1\}^m \rightarrow \{0,1\}$ is defined as $\mathrm{Index}_m(x,y)= y_x$ (the $x$th bit ... more >>>

TR16-070 | 24th April 2016
Mika Göös, Rahul Jain, Thomas Watson

Extension Complexity of Independent Set Polytopes

We exhibit an $n$-node graph whose independent set polytope requires extended formulations of size exponential in $\Omega(n/\log n)$. Previously, no explicit examples of $n$-dimensional $0/1$-polytopes were known with extension complexity larger than exponential in $\Theta(\sqrt{n})$. Our construction is inspired by a relatively little-known connection between extended formulations and (monotone) circuit ... more >>>

TR15-199 | 7th December 2015
Prahladh Harsha, Rahul Jain, Jaikumar Radhakrishnan

Relaxed partition bound is quadratically tight for product distributions

Let $f : \{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}$ be a 2-party function. For every product distribution $\mu$ on $\{0,1\}^n \times \{0,1\}^n$, we show that $${{CC}}^\mu_{0.49}(f) = O\left(\left(\log {{rprt}}_{1/4}(f) \cdot \log \log {{rprt}}_{1/4}(f)\right)^2\right),$$ where ${{CC}^\mu_\varepsilon(f)$ is the distributional communication complexity with error at most $\varepsilon$ under the distribution $\mu$ and ... more >>>

TR15-028 | 27th February 2015
Lila Fontes, Rahul Jain, Iordanis Kerenidis, Sophie Laplante, Mathieu Lauriere, Jérémie Roland

Relative Discrepancy does not separate Information and Communication Complexity

Does the information complexity of a function equal its communication complexity? We examine whether any currently known techniques might be used to show a separation between the two notions. Recently, Ganor et al. provided such a separation in the distributional setting for a specific input distribution ?. We show that ... more >>>

TR13-158 | 18th November 2013
Gábor Braun, Rahul Jain, Troy Lee, Sebastian Pokutta

Information-theoretic approximations of the nonnegative rank

Revisions: 3

Common information was introduced by Wyner as a measure of dependence of two
random variables. This measure has been recently resurrected as a lower bound on the logarithm of the nonnegative rank of a nonnegative matrix. Lower bounds on nonnegative rank have important applications to several areas such
as communication ... more >>>

TR11-033 | 8th March 2011
Rahul Jain, Shengyu Zhang

The influence lower bound via query elimination

We give a simpler proof, via query elimination, of a result due to O'Donnell, Saks, Schramm and Servedio, which shows a lower bound on the zero-error randomized query complexity of a function $f$ in terms of the maximum influence of any variable of $f$. Our lower bound also applies to ... more >>>

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