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 \(2^{\epsilon \cdot n}\), for some \(\epsilon \gt 0\). 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 \(\mathrm{poly}(n)\) can be solved by probabilistic algorithms in time \(2^{n/\mathrm{polylog}(n)}\), then \(\mathcal {BPE}\) does not have circuits of quasilinear size.
中文翻译：
关于指数时间假设、去随机化和电路下界
这指数时间假设 (\(\mathtt {ETH}\))是对\(\mathcal {P} \ne \mathcal {NP}\)猜想的加强，说明\(3\text{-}\mathtt {SAT}\)对\(n\)个变量无法求解在（统一）时间 \(2^{\epsilon \cdot n}\)，对于一些 \(\epsilon \gt 0\)。近年来，其他经典复杂性猜想（如 \(\mathcal {NP}\nsubseteq \mathcal {BPP}\) 或 \(co\mathcal {NP}\nsubseteq \ mathcal {NP}\)) 也被引入，并且具有广泛的影响力。
在这项工作中，我们关注指数时间假设与去随机化和电路下界等基本且密切相关的问题的相互作用。我们表明，即使指数时间假设的相对温和的变体对去随机化、电路下界以及两者之间的联系也具有深远的影响。具体来说，我们证明：
(1) | 这随机指数时间假设 (\(\mathsf {rETH}\))意味着 \(\mathcal {BPP}\) 可以在确定性（几乎）多项式时间（即时间 \(2^{\tilde{O}(\log (n) )}=n^{\mathrm{loglog}(n)^{O(1)}}\))。去随机化依赖于具有近指数拉伸（即种子长度 \(\tilde{O}(\log (n))\)）的伪随机生成器的条件构造；这显着提高了统一“随机难度”结果的最新技术水平，以前只能从此类假设中产生具有次指数拉伸的伪随机生成器。 |
(2) | 这非确定性指数时间假设 (\(\mathsf {NETH}\))意味着 \(\mathcal {BPP}\) 的去随机化完全等同于针对 \(\mathcal {E}\) 的电路下界，特别是伪随机生成器对于去随机化是必要的。事实上，我们证明了上述等价性来自\(\mathsf {NETH}\) 的一个非常弱的版本，并且我们还证明了这个非常弱的版本对于证明我们从中推导出的稍微更强的结论是必要的。 |
最后，我们表明反驳某些指数时间假设需要证明突破电路下限。特别是，如果 \(\mathtt {CircuitSAT}\) 对于大小为 \(\mathrm{poly}(n)\) 的 \(n\) 位电路可以通过概率算法在时间 \(2^ { n /\mathrm{polylog}(n)}\), 那么 \(\mathcal {BPE}\) 没有拟线性大小的电路。
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