Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems

We show that any polygonal subdivision in the plane can be converted into an “mguillotine” subdivision whose length is at most (1 + c m ) times that of the original subdivision, for a small constant c. “m-Guillotine” subdivisions have a simple recursive structure that allows one to search for the shortest of such subdivisions in polynomial time, using dynamic programming. In particular, a consequence of our main theorem is a simple polynomial-time approximation scheme for geometric instances of several network optimization problems, including the Steiner minimum spanning tree, the traveling salesperson problem (TSP), and the k-MST problem.