Polylogarithmic-depth matching sampling

Construct an algorithm that samples the monomer–dimer distribution on every simple graph with polylogarithmic depth and polynomial work at inverse-polynomial total-variation error, preferably with near-linear work.

Background

The paper gives a work-efficient parallel sampler with sublinear depth, including a degree-independent \widetilde O_λ(\sqrt n) bound, but does not achieve polylogarithmic depth on arbitrary graphs.

The stated problem asks for an RNC-type sampler for the full monomer–dimer law, complementing known parallel results for maximum matching. The discussion emphasizes that existing parallel sampling/counting reductions are asymmetric and that known conditional-marginal approaches do not yet yield RNC sampling.

References

Can the monomer--dimer law on every simple graph be sampled with polylogarithmic depth and polynomial work at inverse-polynomial total-variation error, ideally with near-linear work?

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

A concrete goal is to answer $k$ edge-occupation queries in expected $_\lambda(k)$ total online work, uniformly in the maximum degree, where $$ suppresses polylogarithmic factors in~$n$.

— 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”