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Electronic Colloquium on Computational Complexity

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REPORTS > AUTHORS > LIJIE CHEN:
All reports by Author Lijie Chen:

TR22-097 | 3rd July 2022
Lijie Chen, Ron D. Rothblum, Roei Tell

Unstructured Hardness to Average-Case Randomness

The leading technical approach in uniform hardness-to-randomness in the last two decades faced several well-known barriers that caused results to rely on overly strong hardness assumptions, and yet still yield suboptimal conclusions.

In this work we show uniform hardness-to-randomness results that *simultaneously break through all of the known barriers*. Specifically, ... more >>>


TR22-086 | 9th June 2022
Lijie Chen, Jiatu Li, Tianqi Yang

Extremely Efficient Constructions of Hash Functions, with Applications to Hardness Magnification and PRFs

Revisions: 1

In a recent work, Fan, Li, and Yang (STOC 2022) constructed a family of almost-universal hash functions such that each function in the family is computable by $(2n + o(n))$-gate circuits of fan-in $2$ over the $B_2$ basis. Applying this family, they established the existence of pseudorandom functions computable by ... more >>>


TR22-057 | 25th April 2022
Lijie Chen, Roei Tell

When Arthur has Neither Random Coins nor Time to Spare: Superfast Derandomization of Proof Systems

Revisions: 1

What is the actual cost of derandomization? And can we get it for free? These questions were recently raised by Doron et. al (FOCS 2020) and have been attracting considerable interest. In this work we extend the study of these questions to the setting of *derandomizing interactive proofs systems*.

... more >>>

TR21-166 | 21st November 2021
Lijie Chen, Shuichi Hirahara, Neekon Vafa

Average-case Hardness of NP and PH from Worst-case Fine-grained Assumptions

What is a minimal worst-case complexity assumption that implies non-trivial average-case hardness of NP or PH? This question is well motivated by the theory of fine-grained average-case complexity and fine-grained cryptography. In this paper, we show that several standard worst-case complexity assumptions are sufficient to imply non-trivial average-case hardness ... more >>>


TR21-165 | 21st November 2021
Shyan Akmal, Lijie Chen, Ce Jin, Malvika Raj, Ryan Williams

Improved Merlin-Arthur Protocols for Central Problems in Fine-Grained Complexity

Revisions: 1

In a Merlin-Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability $1$, and rejects invalid proofs with probability arbitrarily close to $1$. The running time of such a system is defined to be the length of Merlin's proof plus the running time of Arthur. We ... more >>>


TR21-159 | 15th November 2021
Lijie Chen, Ce Jin, Rahul Santhanam, Ryan Williams

Constructive Separations and Their Consequences

For a complexity class $C$ and language $L$, a constructive separation of $L \notin C$ gives an efficient algorithm (also called a refuter) to find counterexamples (bad inputs) for every $C$-algorithm attempting to decide $L$. We study the questions: Which lower bounds can be made constructive? What are the consequences ... more >>>


TR21-080 | 10th June 2021
Lijie Chen, Roei Tell

Hardness vs Randomness, Revised: Uniform, Non-Black-Box, and Instance-Wise

We propose a new approach to the hardness-to-randomness framework and to the promise-BPP=promise-P conjecture. Classical results rely on non-uniform hardness assumptions to construct derandomization algorithms that work in the worst-case, or rely on uniform hardness assumptions to construct derandomization algorithms that work only in the average-case. In both types of ... 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 >>>


TR21-027 | 24th February 2021
Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh Saxena, Zhao Song, Huacheng Yu

Almost Optimal Super-Constant-Pass Streaming Lower Bounds for Reachability

We give an almost quadratic $n^{2-o(1)}$ lower bound on the space consumption of any $o(\sqrt{\log n})$-pass streaming algorithm solving the (directed) $s$-$t$ reachability problem. This means that any such algorithm must essentially store the entire graph. As corollaries, we obtain almost quadratic space lower bounds for additional fundamental problems, including ... 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 >>>


TR20-148 | 28th September 2020
Lijie Chen, Roei Tell

Simple and fast derandomization from very hard functions: Eliminating randomness at almost no cost

Revisions: 1

Extending the classical ``hardness-to-randomness'' line-of-works, Doron et al. (FOCS 2020) recently proved that derandomization with near-quadratic time overhead is possible, under the assumption that there exists a function in $\mathcal{DTIME}[2^n]$ that cannot be computed by randomized SVN circuits of size $2^{(1-\epsilon)\cdot n}$ for a small $\epsilon$.

In this work we ... more >>>


TR20-065 | 2nd May 2020
Lijie Chen, Ce Jin, Ryan Williams

Sharp Threshold Results for Computational Complexity

We establish several ``sharp threshold'' results for computational complexity. For certain tasks, we can prove a resource lower bound of $n^c$ for $c \geq 1$ (or obtain an efficient circuit-analysis algorithm for $n^c$ size), there is strong intuition that a similar result can be proved for larger functions of $n$, ... more >>>


TR20-010 | 12th February 2020
Lijie Chen, Hanlin Ren

Strong Average-Case Circuit Lower Bounds from Non-trivial Derandomization

Revisions: 1

We prove that for all constants a, NQP = NTIME[n^{polylog(n)}] cannot be (1/2 + 2^{-log^a n})-approximated by 2^{log^a n}-size ACC^0 of THR circuits (ACC^0 circuits with a bottom layer of THR gates). Previously, it was even open whether E^NP can be (1/2+1/sqrt{n})-approximated by AC^0[2] circuits. As a straightforward application, ... more >>>


TR19-169 | 21st November 2019
Lijie Chen, Ron Rothblum, Roei Tell, Eylon Yogev

On Exponential-Time Hypotheses, Derandomization, and Circuit Lower Bounds

Revisions: 2

The Exponential-Time Hypothesis ($ETH$) is a strengthening of the $\mathcal{P} \neq \mathcal{NP}$ conjecture, stating that $3\text{-}SAT$ on $n$ variables cannot be solved in time $2^{\epsilon\cdot n}$, for some $\epsilon>0$. In recent years, analogous hypotheses that are ``exponentially-strong'' forms of other classical complexity conjectures (such as $\mathcal{NP}\not\subseteq\mathcal{BPP}$ or $co\text{-}\mathcal{NP}\not\subseteq \mathcal{NP}$) have ... more >>>


TR19-168 | 20th November 2019
Igor Carboni Oliveira, Lijie Chen, Shuichi Hirahara, Ján Pich, Ninad Rajgopal, Rahul Santhanam

Beyond Natural Proofs: Hardness Magnification and Locality

Hardness magnification reduces major complexity separations (such as $EXP \not\subseteq NC^1$) to proving lower bounds for some natural problem $Q$ against weak circuit models. Several recent works [OS18, MMW19, CT19, OPS19, CMMW19, Oli19, CJW19a] have established results of this form. In the most intriguing cases, the required lower bound is ... more >>>


TR19-118 | 5th September 2019
Lijie Chen, Ce Jin, Ryan Williams

Hardness Magnification for all Sparse NP Languages

In the Minimum Circuit Size Problem (MCSP[s(m)]), we ask if there is a circuit of size s(m) computing a given truth-table of length n = 2^m. Recently, a surprising phenomenon termed as hardness magnification by [Oliveira and Santhanam, FOCS 2018] was discovered for MCSP[s(m)] and the related problem MKtP of ... more >>>


TR19-075 | 25th May 2019
Lijie Chen, Dylan McKay, Cody Murray, Ryan Williams

Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems

Relations and Equivalences Between Circuit Lower Bounds and Karp-Lipton Theorems

A frontier open problem in circuit complexity is to prove P^NP is not in SIZE[n^k] for all k; this is a necessary intermediate step towards NP is not in P/poly. Previously, for several classes containing P^NP, including NP^NP, ZPP^NP, and ... more >>>


TR19-072 | 17th May 2019
Lijie Chen, Ofer Grossman

Broadcast Congested Clique: Planted Cliques and Pseudorandom Generators

Consider the multiparty communication complexity model where there are n processors, each receiving as input a row of an n by n matrix M with entries in {0, 1}, and in each round each party can broadcast a single bit to all other parties (this is known as the BCAST(1) ... more >>>


TR19-031 | 4th March 2019
Lijie Chen

Non-deterministic Quasi-Polynomial Time is Average-case Hard for ACC Circuits

Revisions: 1

Following the seminal work of [Williams, J. ACM 2014], in a recent breakthrough, [Murray and Williams, STOC 2018] proved that NQP (non-deterministic quasi-polynomial time) does not have polynomial-size ACC^0 circuits.

We strengthen the above lower bound to an average case one, by proving that for all constants c, ... more >>>


TR18-199 | 24th November 2018
Lijie Chen, Roei Tell

Bootstrapping Results for Threshold Circuits “Just Beyond” Known Lower Bounds

The best-known lower bounds for the circuit class $\mathcal{TC}^0$ are only slightly super-linear. Similarly, the best-known algorithm for derandomization of this class is an algorithm for quantified derandomization (i.e., a weak type of derandomization) of circuits of slightly super-linear size. In this paper we show that even very mild quantitative ... more >>>


TR18-026 | 7th February 2018
Lijie Chen

On The Hardness of Approximate and Exact (Bichromatic) Maximum Inner Product

Revisions: 1


In this paper we study the (Bichromatic) Maximum Inner Product Problem (Max-IP), in which we are given sets $A$ and $B$ of vectors, and the goal is to find $a \in A$ and $b \in B$ maximizing inner product $a \cdot b$. Max-IP is very basic and serves ... more >>>


TR16-200 | 18th December 2016
Scott Aaronson, Lijie Chen

Complexity-Theoretic Foundations of Quantum Supremacy Experiments

Revisions: 1

In the near future, there will likely be special-purpose quantum computers with 40-50 high-quality qubits. This paper lays general theoretical foundations for how to use such devices to demonstrate "quantum supremacy": that is, a clear quantum speedup for some task, motivated by the goal of overturning the Extended Church-Turing Thesis ... more >>>


TR16-140 | 9th September 2016
Adam Bouland, Lijie Chen, Dhiraj Holden, Justin Thaler, Prashant Nalini Vasudevan

On SZK and PP

Revisions: 3

In both query and communication complexity, we give separations between the class NISZK, containing those problems with non-interactive statistical zero knowledge proof systems, and the class UPP, containing those problems with randomized algorithms with unbounded error. These results significantly improve on earlier query separations of Vereschagin [Ver95] and Aaronson [Aar12] ... more >>>




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