An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures

TL;DR

This paper explores the computational complexity of bending wires and sheet metal into specific shapes, revealing NP-hardness in certain constrained scenarios and providing efficient algorithms for specific manufacturing cases.

Contribution

It analyzes the complexity of wire straightening problems under manufacturing constraints, showing NP-hardness and providing efficient solutions for particular cases.

Findings

Deciding if a linkage with vertex degeneracy can be straightened is NP-hard.

Deciding if a linkage can be straightened with each joint altered once is NP-complete.

An efficient algorithm exists for sequentially straightening a linkage with joint alteration constraints in manufacturing.

Abstract

We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple non-self-intersecting wire structure (a carpenter's ruler) can be straightened, a problem that was open for several years and has only recently been solved in the affirmative. If we impose some of the constraints that are imposed by the manufacturing process, we obtain quite different results. In particular, we study the variant of the carpenter's ruler problem in which there is a restriction that only one joint can be modified at a time. For a linkage that does not self-intersect or self-touch, the recent results of Connelly et al. and Streinu imply that it can always be straightened, modifying one joint at a time. However, we show that for a linkage with even a single vertex degeneracy, it becomes NP-hard to decide if it can be straightened while altering only one joint at a time. If we add the restriction that each joint can be altered at most once, we show that the problem is NP-complete even without vertex degeneracies. In the special case, arising in wire forming manufacturing, that each joint can be altered at most once, and must be done sequentially from one or both ends of the linkage, we give an efficient algorithm to determine if a linkage can be straightened.