Folding One Polyhedral Metric Graph into Another

TL;DR

This paper studies the problem of folding one polyhedral metric graph into another, establishing NP-hardness, optimal scale factors, and complexity results, with applications to Platonic solids and graph transformations.

Contribution

It introduces the NP-hardness of polyhedral graph folding, determines optimal scale factors, and characterizes the problem's computational complexity and rationality constraints.

Findings

Folding between all pairs of Platonic solids is characterized.

The problem is NP-hard and hard to approximate.

Optimal scale factors are always rational, leading to DP-completeness.

Abstract

We analyze the problem of folding one polyhedron, viewed as a metric graph of its edges, into the shape of another, similar to 1D origami. We find such foldings between all pairs of Platonic solids and prove corresponding lower bounds, establishing the optimal scale factor when restricted to integers. Further, we establish that our folding problem is also NP-hard, even if the source graph is a tree. It turns out that the problem is hard to approximate, as we obtain NP-hardness even for determining the existence of a scale factor 1.5-{\epsilon}. Finally, we prove that, in general, the optimal scale factor has to be rational. This insight then immediately results in NP membership. In turn, verifying whether a given scale factor is indeed the smallest possible, requires two independent calls to an NP oracle, rendering the problem DP-complete.