All reports by Author Rahul Ilango:

__
TR23-171
| 15th November 2023
__

Shuichi Hirahara, Rahul Ilango, Ryan Williams#### Beating Brute Force for Compression Problems

__
TR23-165
| 5th November 2023
__

Rahul Ilango#### SAT Reduces to the Minimum Circuit Size Problem with a Random Oracle

__
TR23-046
| 13th April 2023
__

Yizhi Huang, Rahul Ilango, Hanlin Ren#### NP-Hardness of Approximating Meta-Complexity: A Cryptographic Approach

__
TR23-038
| 28th March 2023
__

Rahul Ilango, Jiatu Li, Ryan Williams#### Indistinguishability Obfuscation, Range Avoidance, and Bounded Arithmetic

__
TR23-035
| 22nd March 2023
__

Shuichi Hirahara, Rahul Ilango, Zhenjian Lu, Mikito Nanashima, Igor Carboni Oliveira#### A Duality Between One-Way Functions and Average-Case Symmetry of Information

__
TR21-082
| 16th June 2021
__

Rahul Ilango, Hanlin Ren, Rahul Santhanam#### Hardness on any Samplable Distribution Suffices: New Characterizations of One-Way Functions by Meta-Complexity

__
TR21-030
| 2nd March 2021
__

Shuichi Hirahara, Rahul Ilango, Bruno Loff#### Hardness of Constant-round Communication Complexity

__
TR20-183
| 6th December 2020
__

Rahul Ilango#### Constant Depth Formula and Partial Function Versions of MCSP are Hard

__
TR20-021
| 21st February 2020
__

Rahul Ilango, Bruno Loff, Igor Carboni Oliveira#### NP-Hardness of Circuit Minimization for Multi-Output Functions

__
TR19-021
| 19th February 2019
__

Rahul Ilango#### $AC^0[p]$ Lower Bounds and NP-Hardness for Variants of MCSP

__
TR19-018
| 18th February 2019
__

Alexander Golovnev, Rahul Ilango, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, Avishay Tal#### AC0[p] Lower Bounds against MCSP via the Coin Problem

__
TR18-173
| 17th October 2018
__

Eric Allender, Rahul Ilango, Neekon Vafa#### The Non-Hardness of Approximating Circuit Size

Revisions: 1

Shuichi Hirahara, Rahul Ilango, Ryan Williams

A compression problem is defined with respect to an efficient encoding function $f$; given a string $x$, our task is to find the shortest $y$ such that $f(y) = x$. The obvious brute-force algorithm for solving this compression task on $n$-bit strings runs in time $O(2^{\ell} \cdot t(n))$, where $\ell$ ... more >>>

Rahul Ilango

The Minimum Circuit Size Problem (MCSP) asks, given the truth table of a Boolean function $f$ and an integer $s$, if there is a circuit computing $f$ of size at most $s.$ It has been an open question since Levin's seminal work on NP-completeness whether MCSP is NP-complete. This question ... more >>>

Yizhi Huang, Rahul Ilango, Hanlin Ren

It is a long-standing open problem whether the Minimum Circuit Size Problem ($\mathrm{MCSP}$) and related meta-complexity problems are NP-complete. Even for the rare cases where the NP-hardness of meta-complexity problems are known, we only know very weak hardness of approximation.

In this work, we prove NP-hardness of approximating meta-complexity with ... more >>>

Rahul Ilango, Jiatu Li, Ryan Williams

The range avoidance problem (denoted by Avoid) asks to find a string outside of the range of a given circuit $C:\{0,1\}^n\to\{0,1\}^m$, where $m>n$. Although at least half of the strings of length $m$ are correct answers, it is not clear how to deterministically find one. Recent results of Korten (FOCS'21) ... more >>>

Shuichi Hirahara, Rahul Ilango, Zhenjian Lu, Mikito Nanashima, Igor Carboni Oliveira

Symmetry of Information (SoI) is a fundamental property of Kolmogorov complexity that relates the complexity of a pair of strings and their conditional complexities. Understanding if this property holds in the time-bounded setting is a longstanding open problem. In the nineties, Longpré and Mocas (1993) and Longpré and Watanabe (1995) ... more >>>

Rahul Ilango, Hanlin Ren, Rahul Santhanam

We show that one-way functions exist if and only if there is some samplable distribution D such that it is hard to approximate the Kolmogorov complexity of a string sampled from D. Thus we characterize the existence of one-way functions by the average-case hardness of a natural \emph{uncomputable} problem on ... more >>>

Shuichi Hirahara, Rahul Ilango, Bruno Loff

How difficult is it to compute the communication complexity of a two-argument total Boolean function $f:[N]\times[N]\to\{0,1\}$, when it is given as an $N\times N$ binary matrix? In 2009, Kushilevitz and Weinreb showed that this problem is cryptographically hard, but it is still open whether it is NP-hard.

In this ... more >>>

Rahul Ilango

Attempts to prove the intractability of the Minimum Circuit Size Problem (MCSP) date as far back as the 1950s and are well-motivated by connections to cryptography, learning theory, and average-case complexity. In this work, we make progress, on two fronts, towards showing MCSP is intractable under worst-case assumptions.

While ... more >>>

Rahul Ilango, Bruno Loff, Igor Carboni Oliveira

Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an ... more >>>

Rahul Ilango

The Minimum Circuit Size Problem (MCSP) asks whether a (given) Boolean function has a circuit of at most a (given) size. Despite over a half-century of study, we know relatively little about the computational complexity of MCSP. We do know that questions about the complexity of MCSP have significant ramifications ... more >>>

Alexander Golovnev, Rahul Ilango, Russell Impagliazzo, Valentine Kabanets, Antonina Kolokolova, Avishay Tal

Minimum Circuit Size Problem (MCSP) asks to decide if a given truth table of an $n$-variate boolean function has circuit complexity less than a given parameter $s$. We prove that MCSP is hard for constant-depth circuits with mod $p$ gates, for any prime $p\geq 2$ (the circuit class $AC^0[p])$. Namely, ... more >>>

Eric Allender, Rahul Ilango, Neekon Vafa

The Minimum Circuit Size Problem (MCSP) has been the focus of intense study recently; MCSP is hard for SZK under rather powerful reductions, and is provably not hard under “local” reductions computable in TIME($n^{0.49}$). The question of whether MCSP is NP-hard (or indeed, hard even for small subclasses of P) ... more >>>