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.
