Fine-grained reductions from approximate counting to decision

In this paper, we introduce a general framework for fine-grained reductions of approximate counting problems to their decision versions. (Thus we use an oracle that decides whether any witness exists to multiplicatively approximate the number of witnesses with minimal overhead.) This mirrors a foundational result of Sipser (STOC 1983) and Stockmeyer (SICOMP 1985) in the polynomial-time setting, and a similar result of M\"uller (IWPEC 2006) in the FPT setting. Using our framework, we obtain such reductions for some of the most important problems in fine-grained complexity: the Orthogonal Vectors problem, 3SUM, and the Negative-Weight Triangle problem (which is closely related to All-Pairs Shortest Path). We also provide a fine-grained reduction from approximate #SAT to SAT. Suppose the Strong Exponential Time Hypothesis (SETH) is false, so that for some $1<c<2$ and all $k$ there is an $O(c^n)$-time algorithm for k-SAT. Then we prove that for all $k$, there is an $O((c+o(1))^n)$-time algorithm for approximate #$k$-SAT. In particular, our result implies that the Exponential Time Hypothesis (ETH) is equivalent to the seemingly-weaker statement that there is no algorithm to approximate #3-SAT to within a factor of $1+\epsilon$ in time $2^{o(n)}/\epsilon^2$ (taking $\epsilon > 0$ as part of the input).