a polygonal path with at most k edges, each edge being contained in a face of T and having its endpoints on the boundary of this face. (Our definition is different from the one given by Daescu et al. =-=[5]-=-.) We first give approximation algorithms that return a polygonal path with cost at most (1 + ?) times the cost of any k-link path. (See Section 3.) In the weighted region problem, Mitchell and Papadi