Polyamorous Scheduling (2403.00465v2)

Abstract: Finding schedules for pairwise meetings between the members of a complex social group without creating interpersonal conflict is challenging, especially when different relationships have different needs. We formally define and study the underlying optimisation problem: Polyamorous Scheduling. In Polyamorous Scheduling, we are given an edge-weighted graph and try to find a periodic schedule of matchings in this graph such that the maximal weighted waiting time between consecutive occurrences of the same edge is minimised. We show that the problem is NP-hard and that there is no efficient approximation algorithm with a better ratio than 4/3 unless P = NP. On the positive side, we obtain an $O(\log n)$-approximation algorithm; indeed, a $O(\log \Delta)$-approximation for $\Delta$ the maximum degree, i.e., the largest number of relationships of any individual. We also define a generalisation of density from the Pinwheel Scheduling Problem, "poly density", and ask whether there exists a poly-density threshold similar to the 5/6-density threshold for Pinwheel Scheduling [Kawamura, STOC 2024]. Polyamorous Scheduling is a natural generalisation of Pinwheel Scheduling with respect to its optimisation variant, Bamboo Garden Trimming. Our work contributes the first nontrivial hardness-of-approximation reduction for any periodic scheduling problem, and opens up numerous avenues for further study of Polyamorous Scheduling.

• The paper establishes NP-hardness of the polyamorous scheduling problem and shows it cannot be approximated within a 13/12 ratio unless P=NP.
• It generalizes traditional scheduling problems by modeling periodic meetings as edge-weight constraints, linking to Pinwheel Scheduling and Bamboo Garden Trimming.
• The authors propose an O(log n)-approximation algorithm using edge-coloring and layering techniques to provide predictable performance guarantees.

Polyamorous Scheduling: An Analysis

The paper "Polyamorous Scheduling" by Gąsieniec, Smith, and Wild investigates a novel scheduling problem that generalizes existing graph-based scheduling paradigms. At the core, the paper introduces a scheduling challenge where a periodic meeting schedule needs to be established on an edge-weighted graph, mimicking the dynamics of relationships within a polyamorous community. The objective is to minimize the maximum weighted waiting time for recurring meetings between members, represented as edges in the graph.

Complexity and Hardness Results

One of the significant contributions of the paper is proving that the Polyamorous Scheduling problem is NP-hard. Moreover, the authors show that unless P=NP, the problem cannot be approximated within a ratio better than 13/12. This indication of hardness is achieved through a reduction from the MAX-3SAT problem. It highlights that, even within periodic scheduling problems which traditionally allow some degree of approximation, Polyamorous Scheduling presents unique challenges preventing efficient approximation schemes.

The paper also establishes a strong connection with existing scheduling problems like Pinwheel Scheduling and Bamboo Garden Trimming by defining a generalization of the density threshold. This comparative analysis situates Polyamorous Scheduling well within known combinatorial optimization frameworks, extending known results and paving the way for future research.

Approximation Algorithms

Despite the demonstrated complexity, the paper provides hope for tackling the problem through approximation algorithms. The researchers present an O(log n)-approximation algorithm by effectively leveraging edge colorings and layering techniques. These methods offer practical insights for scenarios where perfect solutions are computationally prohibitive.

A key strategy involves breaking the problem into layers based on edge growth rates and then interleaving them. This innovative approach ensures that even if the problem is hard to solve exactly, reasonably efficient schedules can still be constructed with predictable performance guarantees.

Implications and Future Directions

The implications of this work are substantial, both practically and theoretically. In practical terms, organizations and communities could leverage insights from Polyamorous Scheduling in contexts where periodic interactions need to be managed efficiently, especially when resources (or ‘meetings’) must be distributed across complex networks.

Theoretically, the paper opens multiple avenues for future research, such as exploring more specialized classes of polycules like bipartite graphs and considering extended versions of the problem like fungible or secure polyamorous scheduling. Moreover, the challenge of identifying efficient algorithms remains, particularly in determining whether tighter approximation bounds are achievable for certain graph structures.

Conclusion

In conclusion, "Polyamorous Scheduling" is a rigorous exploration of a complex scheduling problem that blends social dynamics with graph theory. The paper’s contribution to the understanding of NP-hard variants of periodic scheduling is significant, laying a foundational framework for subsequent studies. The discussion around approximation algorithms provides a concrete methodology for practical applications, while the uncovered gaps in complexity and approximation bounds encourage further exploration and refinement within combinatorial scheduling challenges. This research marks an important step in evolving the theory and optimization of periodic scheduling problems.