Counting below the annealing baseline

Determine whether approximate counting of matchings can achieve a polynomial saving over the independent-sample annealing work $\widetilde O_\lambda(mn/\varepsilon^2)$ across all graph densities, and in particular achieve $\widetilde O_\lambda(n^{2-\delta})$ work at constant accuracy on every graph with $m=\Theta(n)$ for some $\delta>0$.

Background

The paper improves the annealing baseline on dense graphs, obtaining \widetilde O_λ(n²/\varepsilon²) work rather than \widetilde O_λ(mn/\varepsilon²). For dense graphs this removes a polynomial factor relative to the baseline, but the improvement is not established uniformly over graph densities.

The open problem asks whether further savings are possible for sparse and intermediate-density graphs. The particularly concrete target is a genuinely subquadratic algorithm for linear-size graphs at constant relative accuracy, potentially using cheap conditional or perfect-marginal samples.

References

Can a polynomial saving over $mn$ be obtained across all graph densities, apart from the input cost? In particular, for every graph with $m=\Theta(n)$, can counting be achieved in $_\lambda(n{2-\delta})$ work at constant accuracy, for some $\delta>0$?

— Sampling Matchings in Near-linear Time  (2609.21936 - Miao et al., 18 Sep 2026) in Section 7, Discussion and open problems, paragraph “Counting below the annealing baseline”