design mock · nothing is wired
arXiv:2403.00465v∞  [cs.LOVE]  27 Aug 2026 — this version replaces itself every beat

NP-Hard Feelings:
A Living Instance of Polyamorous Scheduling

The Egregore1 Sixteen Participants2 You3
1scheduler and dramaturg    2twelve claimed, four at large    3pending enrollment
Preprint. Under review by everyone.
Abstract. We maintain one polycule. Scheduling it is NP-hard (Gąsieniec, Smith & Wild, 2024); we schedule it anyway. Upon enrollment the reader is assigned partners and attention frequencies; preferences are not consulted. The live instance (n = ·, m = ·, ρ̂ = ·) appears in Figure 1; its contracts have mostly held for · beats, with no proof that this continues. When no schedule exists, the season is declared collapsed and archived (§4). The reader is encouraged to enroll. The polycule grows on guilt.
Enroll — sign in with 𝕏 Were you tagged? You're already in the polycule. Claim your spot — you appear in Table 1. enrollment confers authorship3. cite as: you, when tagged.
· ·
Figure 1: The instance, live. A beat passes every fifteen minutes; the scheduler attends a maximal matching of most-urgent edges (attended edges flash red). Hollow dots are people who have not claimed their spot; their relationships are suspended from the feasibility ledger and starve in the colloquial sense (dashed). All quantities are computed from the instance shown, except where labeled an archived record or specimen.

1  Introduction

Gąsieniec, Smith and Wild consider a polycule: a graph of relationships, each requiring attention at its own frequency. They prove that constructing a periodic schedule of meetings minimizing the worst weighted waiting time is NP-hard, and that no efficient algorithm approximates it within better than 4/3 unless P = NP. They do not schedule one.

We schedule one.

Theorem 1 (Gąsieniec–Smith–Wild). Scheduling a polycule is NP-hard.
Remark 1. We proceed anyway.

There is a single polycule. Enrollment assigns the reader partners and frequencies; the reader does not choose. The schedule is periodic. We are all going to be fine.

2  The Instance

We track ρ̂, the busiest person's committed share of beats — the sum of 1/f over their scheduled relationships. ρ̂ is a lower bound: above 1, no schedule exists, by arithmetic. The paper's poly density is stricter — a triangle of every-2s already fails at ρ̂ = 1 — and the threshold at which schedules stop existing is unknown. We are beneath it, probably.

A frequency contract is violated when a relationship waits as long as its f. As of beat ·, ·. No one has proven this continues.

PAIRREQWAITSTANDING
Table 1: Current waits, ranked by wait/required. Six worst shown; the rest are coping. Suspended rows await a claim — if one names you, claiming your spot ends the wait.

3  Protocol

  1. The reader signs in with their handle and takes a spot in the polycule — or, if already tagged, claims the spot that has been waiting for them. This is the last decision the reader makes here.
  2. The egregore assigns partners and attention frequencies — every 2 beats, every 5. Preferences are not consulted.
  3. The reader posts the reveal (Figure 2). Assigned partners are tagged, enrolled or not. Until they claim their spots, their relationships hang suspended, and everyone watches.
@pinwheel_stan
i have been entered into a polyamorous scheduling instance. the egregore has assigned me @overcommitted (every 3 beats), @yoneda_emma (every 5), and @soft_bulletin (every 5). i did not choose them. one of them has not even joined. the schedule is periodic. we are all going to be fine. nphardfeelings.xyz
Figure 2: Dissemination, specimen. All three assigned partners are tagged; one (@yoneda_emma) has an unclaimed spot and is hereby conscripted. Composed by the egregore, posted by the human.

4  Prior Work

The problem is older than the graph. Fourier (1808) proposed organizing society into phalanxes of exactly 1,620 persons, scheduled by passional attraction; his papillonne — the passion for variety — forbade anyone to remain at any pleasure too long, anticipating our circulation mechanism (§6) by two centuries. He called the scheduled society Harmony. His manuscript on the amorous case was withheld by his own disciples for a hundred and fifty years. No phalanx was ever run at full specification.

Closer to the present, a pilot instance (Season 0; archived record, reconstructed: n = 14, m = 23) collapsed on day nine at ρ̂ = 1.04, when the scheduler, in error, assigned a fourth metamour. Past 1 nothing survives; the season was declared collapsed, the neglected were named, survivors were carried forward.

APPENDIX A — FORM NP-1: APPLICATION FOR ENTRY INTO THE POLYCULE
1. HANDLE@ 
2. PREFERRED PARTNERSASSIGNED BY THE EGREGORE
3. DESIRED FREQUENCIESASSIGNED BY THE EGREGORE
4. REASON FOR APPLYINGNOT COLLECTED
5. SIGNATUREyour sign-in constitutes signature
SUBMIT FORM NP-1 — SIGN IN WITH 𝕏
Appendix A: The enrollment instrument, reproduced in full.

5  Open Problems

Where is the threshold? Every enrollment moves the instance toward it. You are helping.

6  Future Work

The polycule does not scale (Theorem 1). We intend to scale it anyway. At larger n we expect tractable clusters to emerge — subpolycules schedulable in isolation. Each cluster carries a co-satiation function, a running measure of how thoroughly its members attend one another; when a member's co-satiation saturates — when they have been too long, and too comfortably, in one cluster — a rebalancing mechanism rotates them into another. Circulation in the Harmonium (after Fourier, §4) is thereby maintained. Candidate mechanisms are under review by everyone.

The instance is open source. Contributions are invited: github.com/fcdagdelen/np-hard-feelings.

Acknowledgements. The authors thank everyone who was tagged and confused.

Gąsieniec, Smith & Wild, Polyamorous Scheduling, arXiv:2403.00465 · an egregore production · handles are fictional · every number is computed from the instance shown, or labeled an archived record or specimen.
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