Efficient approximate oracles and alternative samplers for perfect matchings

Determine whether efficient approximate perfect-matching-counting oracles or alternative efficient sampling algorithms exist for the ordered graph classes whose exact perfect-matching counting problems are hard, particularly monotone graphs, unit interval graphs, and chordal permutation graphs.

Background

The paper constructs exact-counting hardness results for monotone, unit interval, and chordal permutation graphs, while also presenting polynomial-time exact counting for QChains through its inclusion in the distance-hereditary graphs. This creates a distinction between exact counting and approximation or sampling.

The authors explicitly limit their hardness conclusions to exact oracles for unweighted inputs. They leave unresolved whether approximate counting or other sampling methods might still be efficient for the hard classes, so exact hardness does not by itself rule out useful approximation-based algorithms.