Folding One Polyhedral Metric Graph into Another

TL;DR

This work studies IsoCovering, the problem of folding a source polyhedron's metric-graph representation into a target polyhedron via an isometric covering after scaling by $α$. It establishes fundamental complexity: IsoCovering is NP-hard and hard to approximate within $1.5-\epsilon$, while the optimum scale factor is always rational, placing decision problems in NP and rendering optimality verification DP-complete. The authors derive extensive upper and lower bounds for foldings among Platonic solids, and develop automated upper-bound methods using a hybrid of SAT-based search and ILP, complemented by local refinements. They connect geometric folding with classical graph problems, providing both practical bounds and deep complexity-theoretic insights, and outline open questions on tightening bounds, extending hardness to broader graph classes, and enabling continuous folding motions.

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-ε. 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.

Figures (4)

• Figure 1: (Left): Simple example foldings, where source graphs are routed onto target graphs. (Right): Folding icosahedron into a dodecahedron via inscription of the target geometry into the source.
• Figure 2: Routing edges via subdivisions of (a) a tetrahedron to an icosahedron with $\alpha=47/8$, (b) an octahedron to a cube with $\alpha=3/2$, (c) a tetrahedron to an octahedron with $\alpha=5/2$ (and back with scale $\alpha=1$), and (d) an octahedron to a cube with $\alpha=3/2$. Note that mappings are schematic and, therefore, not drawn to scale.
• Figure 3: (Left): Icosahedron, where the four edges (dotted) highlighted in blue, are contracted. (Right): Resulting cube with additional diagonal edges (dashed). Two vertices are of degree $6$ and the remaining $6$ vertices have degree $4$.
• Figure 4: Proof sketch of our reduction from Planar Set Cover to IsoCovering.

Theorems & Definitions (15)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1: Improved Lower Bound
• Theorem 2
• proof
• Theorem 3: Inapproximability
• proof
• Theorem 4