skip to main content


Search for: All records

Creators/Authors contains: "Chen, Lijie"

Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher. Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?

Some links on this page may take you to non-federal websites. Their policies may differ from this site.

  1. The Exponential-Time Hypothesis ( \(\mathtt {ETH} \) ) is a strengthening of the \(\mathcal {P} \ne \mathcal {NP} \) conjecture, stating that \(3\text{-}\mathtt {SAT} \) on n variables cannot be solved in (uniform) time 2 ϵ · n , for some ϵ > 0. In recent years, analogous hypotheses that are “exponentially-strong” forms of other classical complexity conjectures (such as \(\mathcal {NP}\nsubseteq \mathcal {BPP} \) or \(co\mathcal {NP}\nsubseteq \mathcal {NP} \) ) have also been introduced, and have become widely influential. In this work, we focus on the interaction of exponential-time hypotheses with the fundamental and closely-related questions of derandomization and circuit lower bounds . We show that even relatively-mild variants of exponential-time hypotheses have far-reaching implications to derandomization, circuit lower bounds, and the connections between the two. Specifically, we prove that: (1) The Randomized Exponential-Time Hypothesis ( \(\mathsf {rETH} \) ) implies that \(\mathcal {BPP} \) can be simulated on “average-case” in deterministic (nearly-)polynomial-time (i.e., in time \(2^{\tilde{O}(\log (n))}=n^{\mathrm{loglog}(n)^{O(1)}} \) ). The derandomization relies on a conditional construction of a pseudorandom generator with near-exponential stretch (i.e., with seed length \(\tilde{O}(\log (n)) \) ); this significantly improves the state-of-the-art in uniform “hardness-to-randomness” results, which previously only yielded pseudorandom generators with sub-exponential stretch from such hypotheses. (2) The Non-Deterministic Exponential-Time Hypothesis ( \(\mathsf {NETH} \) ) implies that derandomization of \(\mathcal {BPP} \) is completely equivalent to circuit lower bounds against \(\mathcal {E} \) , and in particular that pseudorandom generators are necessary for derandomization. In fact, we show that the foregoing equivalence follows from a very weak version of \(\mathsf {NETH} \) , and we also show that this very weak version is necessary to prove a slightly stronger conclusion that we deduce from it. Lastly, we show that disproving certain exponential-time hypotheses requires proving breakthrough circuit lower bounds. In particular, if \(\mathtt {CircuitSAT} \) for circuits over n bits of size poly( n ) can be solved by probabilistic algorithms in time 2 n /polylog( n ) , then \(\mathcal {BPE} \) does not have circuits of quasilinear size. 
    more » « less
  2. Abstract

    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 provide new Merlin–Arthur proof systems for some key problems in fine-grained complexity. In several cases our proof systems have optimal running time. Our main results include:

    Certifying that a list ofnintegers has no 3-SUM solution can be done in Merlin–Arthur time$$\tilde{O}(n)$$O~(n). Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in$$\tilde{O}(n^{1.5})$$O~(n1.5)time (that is, there is a proof system with proofs of length$$\tilde{O}(n^{1.5})$$O~(n1.5)and a deterministic verifier running in$$\tilde{O}(n^{1.5})$$O~(n1.5)time).

    Counting the number ofk-cliques with total edge weight equal to zero in ann-node graph can be done in Merlin–Arthur time$${\tilde{O}}(n^{\lceil k/2\rceil })$$O~(nk/2)(where$$k\ge 3$$k3). For oddk, this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in anm-edge graph can be done in Merlin–Arthur time$${\tilde{O}}(m)$$O~(m). Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only countk-cliques in unweighted graphs, and had worse running times for smallk.

    Computing the All-Pairs Shortest Distances matrix for ann-node graph can be done in Merlin–Arthur time$$\tilde{O}(n^2)$$O~(n2). Note this is optimal, as the matrix can have$$\Omega (n^2)$$Ω(n2)nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an$$\tilde{O}(n^{2.94})$$O~(n2.94)nondeterministic time algorithm.

    Certifying that ann-variablek-CNF is unsatisfiable can be done in Merlin–Arthur time$$2^{n/2 - n/O(k)}$$2n/2-n/O(k). We also observe an algebrization barrier for the previous$$2^{n/2}\cdot \textrm{poly}(n)$$2n/2·poly(n)-time Merlin–Arthur protocol of R. Williams [CCC’16] for$$\#$$#SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol fork-UNSAT running in$$2^{n/2}/n^{\omega (1)}$$2n/2/nω(1)time. Therefore we have to exploit non-algebrizing properties to obtain our new protocol.

    Certifying a Quantified Boolean Formula is true can be done in Merlin–Arthur time$$2^{4n/5}\cdot \textrm{poly}(n)$$24n/5·poly(n). Previously, the only nontrivial result known along these lines was an Arthur–Merlin–Arthur protocol (where Merlin’s proof depends on some of Arthur’s coins) running in$$2^{2n/3}\cdot \textrm{poly}(n)$$22n/3·poly(n)time.

    Due to the centrality of these problems in fine-grained complexity, our results have consequences for many other problems of interest. For example, our work implies that certifying there is no Subset Sum solution tonintegers can be done in Merlin–Arthur time$$2^{n/3}\cdot \textrm{poly}(n)$$2n/3·poly(n), improving on the previous best protocol by Nederlof [IPL 2017] which took$$2^{0.49991n}\cdot \textrm{poly}(n)$$20.49991n·poly(n)time.

     
    more » « less
  3. Etessami, Kousha ; Feige, Uriel ; Puppis, Gabriele (Ed.)
    We give the first pseudorandom generators with sub-linear seed length for the following variants of read-once branching programs (roBPs): 1) First, we show there is an explicit PRG of seed length O(log²(n/ε)log(n)) fooling unbounded-width unordered permutation branching programs with a single accept state, where n is the length of the program. Previously, [Lee-Pyne-Vadhan RANDOM 2022] gave a PRG with seed length Ω(n) for this class. For the ordered case, [Hoza-Pyne-Vadhan ITCS 2021] gave a PRG with seed length Õ(log n ⋅ log 1/ε). 2) Second, we show there is an explicit PRG fooling unbounded-width unordered regular branching programs with a single accept state with seed length Õ(√{n ⋅ log 1/ε} log 1/ε). Previously, no non-trivial PRG (with seed length less than n) was known for this class (even in the ordered setting). For the ordered case, [Bogdanov-Hoza-Prakriya-Pyne CCC 2022] gave an HSG with seed length Õ(log n ⋅ log 1/ε). 3) Third, we show there is an explicit PRG fooling width w adaptive branching programs with seed length O(log n ⋅ log² (nw/ε)). Here, the branching program can choose an input bit to read depending on its current state, while it is guaranteed that on any input x ∈ {0,1}ⁿ, the branching program reads each input bit exactly once. Previously, no PRG with a non-trivial seed length is known for this class. We remark that there are some functions computable by constant-width adaptive branching programs but not by sub-exponential-width unordered branching programs. In terms of techniques, we indeed show that the Forbes-Kelley PRG (with the right parameters) from [Forbes-Kelley FOCS 2018] already fools all variants of roBPs above. Our proof adds several new ideas to the original analysis of Forbes-Kelly, and we believe it further demonstrates the versatility of the Forbes-Kelley PRG. 
    more » « less