Simple approximation algorithms for Polyamorous Scheduling

TL;DR

The existence of a similar threshold for Polyamorous Scheduling is established and the first non-trivial bounds on the poly density threshold are given, giving the first non-trivial bounds on the poly density threshold.

Abstract

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.

Figures (8)

• Figure 1: An instance of disjoint set of four stars that give the worst-case lower bound of $\Omega(\log n)$ for the maximum heat for an OPS instance when disjoint matching is scheduled.
• Figure 2: A DPS polycule which is schedulable iff there is some assignment that satisfies $(x_1\lor x_2), (\overline{x_1}), (x_1 \lor \overline{x_2} \lor x_3),$ and $(\overline{x_3})$ (which is not possible). Connections between layers are omitted for clarity but flow from top to bottom, starting with the variable layer, then a layer duplicating nodes with frequency 3, the $\mathit{OR}$ layer, and finally the tensioning layer. Note that larger problems will also require a $D_6$ layer to support additional tension gadgets.
• Figure 3: Gadgets for the True Clock and for sample variables $x_1$ and $x_2$ (top, left to right), with shorthand versions shown below. Gadgets are shown connected as they would be in a sample variable layer, and their colours and schedules are discussed in \ref{['sec:TrueClock']} . Further variables can be added to the right.
• Figure 4: Four sample flipper gadgets, each with shorthand versions below them: $\color{applegreen}6_G$ male to $\color{blue-violet}6_P$ female (top left), $\color{blue-violet}6_P$ female to $\color{applegreen}6_G$ male (top right), $\color{blue}3_B$ male to $\color{red}3_R$ female (bottom left), and $\color{red}3_R$ female to $\color{blue}3_B$ male (bottom right). Note that $F_3$ flippers act incidentally as $F_6$ flippers, but are used separately in our construction.
• Figure 5: A gadget for duplicating input edges with frequency 3, with a shorthand version on the right. Note that the input can be duplicated indefinitely many times by adding further layers which each use the $9_{b}$ edges from the previous layer, but the output will alternate between male and female $3_a$ edges. Odd layers will have female main nodes with $F_6$ flipper nodes on the right, as shown in the top layer, while even layers will have male main nodes with $F_6$ flipper nodes on the left, as shown on the bottom layer. $\color{applegreen}6_P$ edges can be connected continuously from layer to layer, with the final edge released to another gadget.

Theorems & Definitions (64)

• Theorem 1.1: Poly Density Approximation
• Theorem 1.2: Density Threshold
• Theorem 1.3: SAT Hardness of approximation
• Definition 2.1: Optimisation Polyamorous Scheduling
• Definition 2.2: Decision Polyamorous Scheduling
• Lemma 2.3: OPS to DPS, GasieniecSmithWild2024
• Lemma 2.4: DPS to OPS, GasieniecSmithWild2024
• Definition 3.1: Reduce-Fastest$(x)$ for OPS
• Theorem 3.2
• proof