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REPORTS > KEYWORD > MINIMUM CIRCUIT SIZE PROBLEM:
Reports tagged with Minimum Circuit Size Problem:
TR14-068 | 5th May 2014
Eric Allender, Bireswar Das

#### Zero Knowledge and Circuit Minimization

Revisions: 1

We show that every problem in the complexity class SZK (Statistical Zero Knowledge) is
efficiently reducible to the Minimum Circuit Size Problem (MCSP). In particular Graph Isomorphism lies in RP^MCSP.

This is the first theorem relating the computational power of Graph Isomorphism and MCSP, despite the long history these ... more >>>

TR14-164 | 30th November 2014
Cody Murray, Ryan Williams

#### On the (Non) NP-Hardness of Computing Circuit Complexity

The Minimum Circuit Size Problem (MCSP) is: given the truth table of a Boolean function $f$ and a size parameter $k$, is the circuit complexity of $f$ at most $k$? This is the definitive problem of circuit synthesis, and it has been studied since the 1950s. Unlike many problems of ... more >>>

TR14-176 | 16th December 2014
Eric Allender, Dhiraj Holden, Valentine Kabanets

#### The Minimum Oracle Circuit Size Problem

We consider variants of the Minimum Circuit Size Problem MCSP, where the goal is to minimize the size of oracle circuits computing a given function. When the oracle is QBF, the resulting problem MCSP$^{QBF}$ is known to be complete for PSPACE under ZPP reductions. We show that it is not ... more >>>

TR15-162 | 9th October 2015
Eric Allender, Joshua Grochow, Cris Moore

#### Graph Isomorphism and Circuit Size

Revisions: 1

We show that the Graph Automorphism problem is ZPP-reducible to MKTP, the problem of minimizing time-bounded Kolmogorov complexity. MKTP has previously been studied in connection with the Minimum Circuit Size Problem (MCSP) and is often viewed as essentially a different encoding of MCSP. All prior reductions to MCSP have applied ... more >>>

TR15-198 | 30th November 2015
Shuichi Hirahara, Osamu Watanabe

#### Limits of Minimum Circuit Size Problem as Oracle

Revisions: 1

The Minimum Circuit Size Problem (MCSP) is known to be hard for statistical zero knowledge via a BPP-reduction (Allender and Das, 2014), whereas establishing NP-hardness of MCSP via a polynomial-time many-one reduction is difficult (Murray and Williams, 2015) in the sense that it implies ZPP $\neq$ EXP, which is a ... more >>>

TR16-108 | 16th July 2016
Michael Rudow

#### Discrete Logarithm and Minimum Circuit Size

This paper shows that the Discrete Logarithm Problem is in ZPP^(MCSP) (where MCSP is the Minimum Circuit Size Problem). This result improves the previous bound that the Discrete Logarithm Problem is in BPP^(MCSP) Allender et al. (2006). In doing so, this paper helps classify the relative difficulty of the Minimum ... more >>>

TR16-197 | 7th December 2016
Igor Carboni Oliveira, Rahul Santhanam

#### Conspiracies between Learning Algorithms, Circuit Lower Bounds and Pseudorandomness

We prove several results giving new and stronger connections between learning theory, circuit complexity and pseudorandomness. Let C be any typical class of Boolean circuits, and C[s(n)] denote n-variable C-circuits of size at most s(n). We show:

Learning Speedups: If C[$n^{O(1)}$] admits a randomized weak learning algorithm under the uniform ... more >>>

TR17-158 | 23rd October 2017
Eric Allender, Joshua Grochow, Dieter van Melkebeek, Cris Moore, Andrew Morgan

#### Minimum Circuit Size, Graph Isomorphism, and Related Problems

We study the computational power of deciding whether a given truth-table can be described by a circuit of a given size (the Minimum Circuit Size Problem, or MCSP for short), and of the variant denoted as MKTP where circuit size is replaced by a polynomially-related Kolmogorov measure. All prior reductions ... more >>>

TR18-030 | 9th February 2018
Shuichi Hirahara, Igor Carboni Oliveira, Rahul Santhanam

#### NP-hardness of Minimum Circuit Size Problem for OR-AND-MOD Circuits

The Minimum Circuit Size Problem (MCSP) asks for the size of the smallest boolean circuit that computes a given truth table. It is a prominent problem in NP that is believed to be hard, but for which no proof of NP-hardness has been found. A significant number of works have ... more >>>

TR18-138 | 10th August 2018
Shuichi Hirahara

#### Non-black-box Worst-case to Average-case Reductions within NP

Revisions: 1

There are significant obstacles to establishing an equivalence between the worst-case and average-case hardness of NP: Several results suggest that black-box worst-case to average-case reductions are not likely to be used for reducing any worst-case problem outside coNP to a distributional NP problem.

This paper overcomes the barrier. We ... more >>>

TR18-173 | 17th October 2018
Eric Allender, Rahul Ilango, Neekon Vafa

#### The Non-Hardness of Approximating Circuit Size

Revisions: 1

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 >>>

TR19-155 | 6th November 2019
Rahul Santhanam

#### Pseudorandomness and the Minimum Circuit Size Problem

We explore the possibility of basing one-way functions on the average-case hardness of the fundamental Minimum Circuit Size Problem (MCSP[$s$]), which asks whether a Boolean function on $n$ bits specified by its truth table has circuits of size $s(n)$.

1. (Pseudorandomness from Zero-Error Average-Case Hardness) We show that for ... more >>>

TR20-021 | 21st February 2020
Rahul Ilango, Bruno Loff, Igor Carboni Oliveira

#### NP-Hardness of Circuit Minimization for Multi-Output Functions

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 >>>

TR20-103 | 9th July 2020
Mahdi Cheraghchi, Shuichi Hirahara, Dimitrios Myrisiotis, Yuichi Yoshida

#### One-Tape Turing Machine and Branching Program Lower Bounds for MCSP

Revisions: 1

For a size parameter $s\colon\mathbb{N}\to\mathbb{N}$, the Minimum Circuit Size Problem (denoted by ${\rm MCSP}[s(n)]$) is the problem of deciding whether the minimum circuit size of a given function $f \colon \{0,1\}^n \to \{0,1\}$ (represented by a string of length $N := 2^n$) is at most a threshold $s(n)$. A ... more >>>

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