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.

翻译：我们分析了将一个多面体（视作其边构成的度量图）折叠成另一个多面体形状的问题，类似于一维折纸。我们找到了所有柏拉图立体之间的此类折叠方式，并证明了相应的下界，从而确定了在限制为整数时的最优缩放因子。此外，我们证明了即使源图是树结构，该折叠问题也是NP难的。结果表明，该问题难以近似，因为即使对于确定是否存在1.5-{\epsilon}的缩放因子，我们也证明了其NP难性。最后，我们证明了在一般情况下，最优缩放因子必须是有理数。这一洞见随即直接推导出该问题属于NP类。反过来，验证给定缩放因子是否确实为最小可能值，需要两次独立调用NP预言机，使得该问题成为DP完全问题。