Faster Exponential-Time Approximate Counting via Bounded Self-Reductions

We give faster exponential-time randomised approximation algorithms for counting problems where polynomial-time approximation is unavailable and exact exponential-time counting remains expensive. For general $n$-vertex graphs, our independent-set counter runs in $O^{\ast}(1.1869^{n})$ time, improving the previous $O^{\ast}(1.2041^{n})$ general-graph bound. For $n$-variable \#\textsc{2-SAT}, we obtain an $O^{\ast}(1.2373^{n})$-time approximation algorithm, narrowly below Wahlstr{\"o}m's currently cited $O^{\ast}(1.2377^{n})$ variable-parameter exact bound. The new algorithmic point is to take the square root after decomposition. For a single bounded unweighted self-reduction with $f(x)$ positive leaves and recursion-compatible upper bound $b(x)$, an enumerate-or-sample estimator gives an $(\varepsilon,\delta)$-approximation in $$O^{\ast}\!\left(\sqrt{b(x)}\,\varepsilon^{-2}\log \tfrac1\delta\right)$$ time. After preprocessing decomposes an input into many bounded cores, the combined estimator pays $$O^{\ast}\!\left(\sqrt{\sum_i b_i(x_i)}\,\varepsilon^{-2}\log \tfrac1\delta\right),$$ rather than estimating the cores separately at cost $\sum_i \sqrt{b_i(x_i)}$. The same conversion improves the bases for counting maximal cliques, minimal separators, and perfect matchings in subcubic graphs. Bounded unweighted self-reductions provide the formal language; at the level of counting classes, the resulting unweighted formulation has the same Karp closure as TotP. With explicit recursion-tree access, the framework yields black-box quantum speed-ups.