Computing the Fréchet Distance between Folded Polygons

We present the first results showing that the Fréchet distance between non-flat surfaces can be approximated within a constant factor in polynomial time. Computing the Fréchet distance for surfaces is a surprisingly hard problem. It is not known whether it is computable, it has been shown to be NP-hard, and the only known algorithm computes the Fréchet distance for flat surfaces (Buchin et al.). We adapt this algorithm to create one for computing the Fréchet distance for a class of surfaces which we call folded polygons. Unfortunately, if extended directly the original algorithm no longer guarantees that a homeomorphism exists between the surfaces. We present three different methods to address this problem. The first of which is a fixed-parameter tractable algorithm. The second is a polynomial-time approximation algorithm which approximates the optimum mapping. Finally, we present a restricted class of folded polygons for which we can compute the Fréchet distance in polynomial time.