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Electronic Colloquium on Computational Complexity

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All reports by Author Xin Lyu:

TR22-102 | 15th July 2022
Venkatesan Guruswami, Xin Lyu, Xiuhan Wang

Range Avoidance for Low-depth Circuits and Connections to Pseudorandomness

In the range avoidance problem, the input is a multi-output Boolean circuit with more outputs than inputs, and the goal is to find a string outside its range (which is guaranteed to exist). We show that well-known explicit construction questions such as finding binary linear codes achieving the Gilbert-Varshamov bound ... more >>>

TR22-021 | 19th February 2022
Xin Lyu

Improved Pseudorandom Generators for $\mathrm{AC}^0$ Circuits

We give PRG for depth-$d$, size-$m$ $\mathrm{AC}^0$ circuits with seed length $O(\log^{d-1}(m)\log(m/\varepsilon)\log\log(m))$. Our PRG improves on previous work [TX13, ST19, Kel21] from various aspects. It has optimal dependence on $\frac{1}{\varepsilon}$ and is only one “$\log\log(m)$” away from the lower bound barrier. For the case of $d=2$, the seed length tightly ... more >>>

TR21-040 | 15th March 2021
Lijie Chen, Zhenjian Lu, Xin Lyu, Igor Carboni Oliveira

Majority vs. Approximate Linear Sum and Average-Case Complexity Below NC1

We develop a general framework that characterizes strong average-case lower bounds against circuit classes $\mathcal{C}$ contained in $\mathrm{NC}^1$, such as $\mathrm{AC}^0[\oplus]$ and $\mathrm{ACC}^0$. We apply this framework to show:

- Generic seed reduction: Pseudorandom generators (PRGs) against $\mathcal{C}$ of seed length $\leq n -1$ and error $\varepsilon(n) = n^{-\omega(1)}$ can ... more >>>

TR21-003 | 6th January 2021
Lijie Chen, Xin Lyu

Inverse-Exponential Correlation Bounds and Extremely Rigid Matrices from a New Derandomized XOR Lemma

In this work we prove that there is a function $f \in \textrm{E}^\textrm{NP}$ such that, for every sufficiently large $n$ and $d = \sqrt{n}/\log n$, $f_n$ ($f$ restricted to $n$-bit inputs) cannot be $(1/2 + 2^{-d})$-approximated by $\textrm{F}_2$-polynomials of degree $d$. We also observe that a minor improvement ... more >>>

TR20-150 | 7th October 2020
Lijie Chen, Xin Lyu, Ryan Williams

Almost-Everywhere Circuit Lower Bounds from Non-Trivial Derandomization

In certain complexity-theoretic settings, it is notoriously difficult to prove complexity separations which hold almost everywhere, i.e., for all but finitely many input lengths. For example, a classical open question is whether $\mathrm{NEXP} \subset \mathrm{i.o.-}\mathrm{NP}$; that is, it is open whether nondeterministic exponential time computations can be simulated on infinitely ... more >>>

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