Simple approximation algorithms for Polyamorous Scheduling

In Polyamorous Scheduling, we are given an edge-weighted graph and must find a periodic schedule of matchings in this graph which minimizes the maximal weighted waiting time between consecutive occurrences of the same edge. This NP-hard problem generalises Bamboo Garden Trimming and is motivated by the need to find schedules of pairwise meetings in a complex social group. We present two different analyses of an approximation algorithm based on the Reduce-Fastest heuristic, from which we obtain first a 6-approximation and then a 5.24-approximation for Polyamorous Scheduling. We also strengthen the extant proof that there is no polynomial-time $(1+δ)$-approximation algorithm for the Optimisation Polyamorous Scheduling problem for any $δ< \frac1{12}$ unless P = NP to the bipartite case. The decision version of Polyamorous Scheduling has a notion of density, similar to that of Pinwheel Scheduling, where problems with density below the threshold are guaranteed to admit a schedule (cf. the recently proven 5/6 conjecture, Kawamura, STOC 2024). We establish the existence of a similar threshold for Polyamorous Scheduling and give the first non-trivial bounds on the poly density threshold.