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^{∗}(1.1869ⁿ) time, improving the previous O^*(1.2041ⁿ) general-graph bound. For n-variable #2-SAT, we obtain an O^*(1.2373ⁿ)-time approximation algorithm, narrowly below Wahlström’s currently cited O^*(1.2377ⁿ) 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 (ε,δ)-approximation in O^*(√{b(x)} ε^{-2}log(1/δ)) time. After preprocessing decomposes an input into many bounded cores, the combined estimator pays O^*(√{∑_i b_i(x_i)} ε^{-2} log (1/δ)) , rather than estimating the cores separately at cost ∑_i √{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.