Computing the Fréchet distance with shortcuts is NP-hard

Abstract: We study the shortcut Fr\'{e}chet distance, a natural variant of the Fr\'{e}chet distance, that allows us to take shortcuts from and to any point along one of the curves. The classic Fr\'echet distance is a bottle-neck distance measure and hence quite sensitive to outliers. The shortcut Fr\'{e}chet distance allows us to cut across outliers and hence produces significantly more meaningful results when dealing with real world data. Driemel and Har-Peled recently described approximation algorithms for the restricted case where shortcuts have to start and end at input vertices. We show that, in the general case, the problem of computing the shortcut Fr\'{e}chet distance is NP-hard. This is the first hardness result for a variant of the Fr\'{e}chet distance between two polygonal curves in the plane. We also present two algorithms for the decision problem: a 3-approximation algorithm for the general case and an exact algorithm for the vertex-restricted case. Both algorithms run in O(n^3 log n) time.