The complexity of pinning simple multiloops

TL;DR

This paper investigates the computational complexity of pinning simple multiloops on surfaces, showing polynomial-time solvability for up to three strands and NP-completeness for twenty or more strands.

Contribution

It establishes the complexity thresholds for the pinning problem on simple multiloops in fixed surfaces, revealing a transition from P to NP-complete as the number of strands increases.

Findings

Problem is in P for s ≤ 3 strands.

Problem is NP-complete for s ≥ 20 strands.

Identifies a complexity transition point for simple multiloops.

Abstract

A multiloop with $s\in \mathbb{N}$ strands is a generic immersion $\gamma\colon \sqcup_1^s \mathbb{S}^1 \looparrowright \Sigma$ of the union of $s$ circles into a surface $\Sigma$, considered up to homeomorphisms. A pinning set of $\gamma$ is a set of points $P\subset \Sigma\setminus \operatorname{im}(\gamma)$, such that in the punctured surface $\Sigma \setminus P$, the immersion $\gamma$ has the minimal number of double points in its homotopy class. Its pinning number $\varpi(\gamma)$ is the minimum cardinal of its pinning sets. In any fixed orientable surface $\Sigma$, the pinning problem which given a multiloop $\gamma$ and $k\in \mathbb{N}$ decides whether $\varpi(\gamma)\le k$ 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 $\Sigma$, the problem is in P when $s\leq 3$ and NP-complete when $s\geq 20$, and present some follow-up questions and conjectures.