Poly Density Threshold

Establish whether there exists a universal constant c such that every Decision Polyamorous Scheduling instance Pa = (P, R, f), whose poly density h*(Pa) is defined via the dual linear program in Theorem 1.7 applied to the corresponding Optimisation instance (P, R, 1/f), admits a valid periodic schedule whenever h*(Pa) ≤ c.

Background

The paper introduces a fractional relaxation of Optimisation Polyamorous Scheduling and derives a dual linear program whose optimal value serves as an instance-specific lower bound termed poly density. This generalizes the notion of density from Pinwheel Scheduling in a way that captures the richer structure of matchings in general graphs.

For Pinwheel Scheduling, a sharp threshold on instance density guarantees schedulability. The authors ask whether a similar universal threshold exists for Polyamorous Scheduling when density is defined through their dual LP (Theorem 1.7), thereby providing a simple feasibility criterion across all instances.