Faster Exponential-Time Approximate Counting via Bounded Self-Reductions

2026-07-07

AI Summary

This work addresses #P-hard counting problems—such as counting independent sets in general graphs and #2-SAT—that are inapproximable in polynomial time and prohibitively expensive to solve exactly. The authors propose a novel framework based on bounded, unweighted self-reducibility, which recursively decomposes problem instances and aggregates upper bounds from subproblems at a square-root recursion depth. By integrating enumeration with a hybrid sampling estimator, the approach substantially reduces the base of the exponential time complexity. The method achieves improved runtimes of O*(1.1869ⁿ) for independent set counting and O*(1.2373ⁿ) for #2-SAT approximation, outperforming the best known exact algorithms. It further extends to counting maximum cliques, minimal separators, and perfect matchings in subcubic graphs, and admits black-box quantum speedup.

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{ö}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 \((\varepsilon,δ)\)-approximation in \[ O^{\ast}\!\left(\sqrt{b(x)}\,\varepsilon^{-2}\log \tfrac1δ\right) \] time. After preprocessing decomposes an input into many bounded cores, the combined estimator pays \[ O^{\ast}\!\left(\sqrt{\sum_i b_i(x_i)}\,\varepsilon^{-2}\log \tfrac1δ\right), \] 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.

Problem

• approximate counting
• exponential-time algorithms
• self-reductions
• randomized algorithms
• counting problems

Innovation

• bounded self-reduction
• exponential-time approximation
• randomized counting
• square-root speedup
• TotP