The complexity of pinning simple multiloops

TL;DR

This work analyzes the computational complexity of pinning simple multiloops on surfaces, revealing a phase transition: the pinning problem SimplePin(Σ,s) is solvable in polynomial time for up to three strands but becomes NP-complete once the strand count reaches twenty (for fixed orientable Σ). Central to the approach is encoding pinning as the satisfaction of a positive CNF (the mobidisc formula) and reducing pinning questions to Vertex Cover variants; the authors provide a detailed 20-strand construction via edge gadget bundles and planar embeddings with a limited slope set to achieve NP-hardness. They also establish tractability for small s through bipartite-vertex-cover reductions and discuss broader directions, including non-orientable surfaces, SAT-based heuristics, random models, and combinatorial pinning games. Overall, the paper maps a sharp complexity boundary for pinning simple multiloops and lays groundwork for deeper study of average-case behavior and related games in topological settings.

Abstract

A multiloop with $s\in \mathbb{N}$ strands is a generic immersion $γ\colon \sqcup_1^s \mathbb{S}^1 \looparrowright Σ$ of the union of $s$ circles into a surface $Σ$, considered up to homeomorphisms. A pinning set of $γ$ is a set of points $P\subset Σ\setminus \operatorname{im}(γ)$, such that in the punctured surface $Σ\setminus P$, the immersion $γ$ has the minimal number of double points in its homotopy class. Its pinning number $\varpi(γ)$ is the minimum cardinal of its pinning sets. In any fixed orientable surface $Σ$, the pinning problem which given a multiloop $γ$ and $k\in \mathbb{N}$ decides whether $\varpi(γ)\le k$ has been show to be NP-complete, even in restrictions to loops (with $s=1$ strand). In this work we study the complexity of the pinning problem in restriction to multiloops whose strands are simple (embedded circles). We show that in any fixed oriented surface $Σ$, the problem is in P when $s\leq 3$ and NP-complete when $s\geq 20$, and present some follow-up questions and conjectures.

Figures (22)

• Figure 1: A planar simple multiloop with $3$ strands, drawn with a tool created by Ryan Pham and the second and third author using a circle-packing algorithm from PADS collins_stephenson_2003eppsteinPADSPham_Computing_mobidisc_formulae_2026.
• Figure 2: Left: The simple multiloop https://christopherlloyd.github.io/LooPindex/multiloops/12%5E3_56.html has number pinning number $5$, two optimal pinning sets (denoted by capital letters), and seven other minimal pinning sets of size $6$ (denoted by lowercase letters). Right: The part of its pinning ideal generated by unions of minimal pinning sets.
• Figure 3: Examples and non-examples of different kinds of monorbigons.
• Figure 4: This multiloop $\mathbb{S}^1 \sqcup \mathbb{S}^1 \looparrowright \mathbb{S}^2\setminus \{p_1,p_2\}$ is not taut yet has no singular monorbigons.
• Figure 5: Left: Local strands around an edge. Right: A $4$-colored multiloop associated to the $1$-skeleton of the pentagonal prism.

Theorems & Definitions (59)

• Definition 1: multiloop, simple multiloop, pinning set, pinning number
• Remark 2: no orientations
• Example 3: empirical structure of pinning ideals
• Definition 4: pinning problem
• Remark 5: filling multiloops
• Definition 6: combinatorial encodings of multiloops
• Definition 7: $\mathsf{SimplePin}$
• Theorem 8: in P for $\le 3$ strands
• proof : Proof outline
• Theorem 9: NP-complete for many strands