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Electronic Colloquium on Computational Complexity

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Reports tagged with meta-complexity:
TR20-143 | 16th September 2020
Shuichi Hirahara

Characterizing Average-Case Complexity of PH by Worst-Case Meta-Complexity

We exactly characterize the average-case complexity of the polynomial-time hierarchy (PH) by the worst-case (meta-)complexity of GapMINKT(PH), i.e., an approximation version of the problem of determining if a given string can be compressed to a short PH-oracle efficient program. Specifically, we establish the following equivalence:

DistPH is contained in ... more >>>

TR21-030 | 2nd March 2021
Shuichi Hirahara, Rahul Ilango, Bruno Loff

Hardness of Constant-round Communication Complexity

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 >>>

TR21-058 | 21st April 2021
Shuichi Hirahara

Average-Case Hardness of NP from Exponential Worst-Case Hardness Assumptions

A long-standing and central open question in the theory of average-case complexity is to base average-case hardness of NP on worst-case hardness of NP. A frontier question along this line is to prove that PH is hard on average if UP requires (sub-)exponential worst-case complexity. The difficulty of resolving this ... more >>>

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

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 >>>

TR21-089 | 25th June 2021
Hanlin Ren, Rahul Santhanam

A Relativization Perspective on Meta-Complexity

Meta-complexity studies the complexity of computational problems about complexity theory, such as the Minimum Circuit Size Problem (MCSP) and its variants. We show that a relativization barrier applies to many important open questions in meta-complexity. We give relativized worlds where:

* MCSP can be solved in deterministic polynomial time, but ... more >>>

TR22-023 | 19th February 2022
Erfan Khaniki

Nisan--Wigderson generators in Proof Complexity: New lower bounds

A map $g:\{0,1\}^n\to\{0,1\}^m$ ($m>n$) is a hard proof complexity generator for a proof system $P$ iff for every string $b\in\{0,1\}^m\setminus Rng(g)$, formula $\tau_b(g)$ naturally expressing $b\not\in Rng(g)$ requires superpolynomial size $P$-proofs. One of the well-studied maps in the theory of proof complexity generators is Nisan--Wigderson generator. Razborov (Annals of Mathematics ... more >>>

TR22-072 | 15th May 2022
Halley Goldberg, Valentine Kabanets, Zhenjian Lu, Igor Oliveira

Probabilistic Kolmogorov Complexity with Applications to Average-Case Complexity

Understanding the relationship between the worst-case and average-case complexities of $\mathrm{NP}$ and of other subclasses of $\mathrm{PH}$ is a long-standing problem in complexity theory. Over the last few years, much progress has been achieved in this front through the investigation of meta-complexity: the complexity of problems that refer to the ... more >>>

TR22-081 | 26th May 2022
Zhenjian Lu, Igor Oliveira

Theory and Applications of Probabilistic Kolmogorov Complexity

Diverse applications of Kolmogorov complexity to learning [CIKK16], circuit complexity [OPS19], cryptography [LP20], average-case complexity [Hir21], and proof search [Kra22] have been discovered in recent years. Since the running time of algorithms is a key resource in these fields, it is crucial in the corresponding arguments to consider time-bounded variants ... more >>>

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