Folding Links is Hard

The ruler folding problem asks: Given a set of n line segment links attached at their end points with rotating joints. Find a minimum length folding of this chain of links along a line. In this paper, we provide an improved approximation for the problem if the longest link is significantly larger than the rest. We then consider generalizations to trees of links and instances containing cycles of links. We provide the first fully polynomial time approximation scheme (FPTAS) for the tree variant and prove the inapproximability of cycle variants. Lastly, we consider the area optimizaton problem, of folding the links to achieve minimum area within minimum width. We prove the problem and any multiplicative approximation to be NP-hard and also prove the impossibility of any additive polynomial time approximation schemes (PTAS).