Polynomial-activity matching sampling

Construct a monomer–dimer sampler whose running time is polynomial in the graph size, \log\lambda, and 1/\varepsilon for activities \lambda\ge1, thereby enabling approximate uniform sampling of perfect matchings through sufficiently large activities.

Background

The paper proves near-linear graph-size dependence for fixed activity, but its constants may have unfavorable dependence on the activity. The discussion first asks whether near-linear graph-size dependence can be retained with only polynomial dependence on λ.

It then formulates a stronger goal: dependence polynomial in n, m, \log\lambda, and 1/\varepsilon. Such a result would address approximate uniform sampling of perfect matchings because sufficiently large activity concentrates the monomer–dimer distribution on maximum-cardinality matchings. The authors note that this goal requires ideas beyond merely sharpening worst-start mixing bounds for single-edge Glauber dynamics.

References

Can our Glauber mixing and sampling bounds retain near-linear dependence on graph size with only polynomial dependence on $\lambda$? A more ambitious goal is a monomer--dimer sampler whose running time, for $\lambda\ge1$, is polynomial in $n,m,\log\lambda$, and $1/\varepsilon$.

— Sampling Matchings in Near-linear Time  (2609.21936 - Miao et al., 18 Sep 2026) in Section 7, Discussion and open problems, paragraph “Activity dependence and the perfect-matching limit”

Can one generate $M\sim\mu_{G,\lambda}$ exactly in expected $_\lambda(m+n)$ time?

— Sampling Matchings in Near-linear Time  (2609.21936 - Miao et al., 18 Sep 2026) in Section 7, Discussion and open problems, paragraph “Exact and local sampling of matchings”