Faster Exponential-Time Approximate Counting via Bounded Self-Reductions

Abstract

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.