Angle-Restricted Tours in the Plane

For a given set A C_ (-Tr; +Tr] of angles, the problem "Angle-Restricted Tour" (ART) is to decide whether a set P of n points in the Euclidean plane allows a closed directed tour consisting of straight line segments, such that all angles between consecutive line segments are from the set A. We present a variety of algorithmic and combinatorial results on this problem. In particular, we show that any finite set of at least five points allows a "pseudoconvex" tour (i.e., a tour where all angles are nonnegative), and we derive a fast algorithm for constructing such a tour. Moreover, we give a complete classification (from the computational complexity point of view) for the special cases where the tour has to be part of the orthogonal grid. © 1997 Elsevier Science B.V.