ETH is Dead, Long Live SH

Abstract

Impagliazzo and Paturi's 1999 Exponential Time Hypothesis (ETH) is false, but Stearns and Hunt's 1990 Satisfiability Hypothesis (SH) remains plausible. We prove that SH is equivalent to the recently proposed Weak ETH which rules out $O(2^{n^{1-\varepsilon}})L^{O(1)}$-time 3SAT algorithms for every constant $\varepsilon>0$, where $L$ is the input length. We show that SH (or Weak ETH) preserves the typical process for lower bounds: a polynomial-time reduction from a 3SAT formula of size $s=Θ(n+m)$ to an instance of size $N\le s^{b+o(1)}$, for fixed blowup exponent $b>0$, rules out $O(2^{N^{1/b-\varepsilon}})$-time algorithms for every constant $\varepsilon>0$. By comparison, for an $N=O(s^b)$ blowup, ETH would have ruled out $2^{o(N^{1/b})}$-time algorithms. The framework also yields parameterized lower bounds ruling out running times of the form $O(2^{k^{1/b-\varepsilon}})L^{O(1)}$. We also prove equivalence of the analogous counting hypotheses, #SH and Weak #ETH, and that they rule out $O(2^{(n+m)^{1-\varepsilon}})$-time algorithms for #2SAT for every constant $\varepsilon>0$.