Abstract. 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. Key words. approximation algorithms, polynomial-time approximation scheme, traveling salesperson problem, k-MST, Steiner minimal trees, guillotine subdivisions, computational geometry AMS subject classifications. 68Q25, 68R10, 68U05 PII. S0097539796309764 1. Introduction. We obtain a simple polynomial-time approximation scheme for geometric instances of some network optimization problems, including the Steiner minimum spanning tree, the traveling salesperson problem (TSP), and the k-MST problem. The method is based on the concept of an "m-guillotine subdivision," a simple extension of the recent approximation method of Mitchell [9] , which considered the case m = 1. Roughly speaking, an "m-guillotine subdivision" is a polygonal subdivision with the property that there exists a line ("cut"), whose intersection with the subdivision edges consists of a small number (O(m)) of connected components, and the subdivisions on either side of the line are also m-guillotine. The upper bound on the number of connected components allows one to apply dynamic programming to optimize over m-guillotine subdivisions, as there is a succinct specification of how subproblems interact across a cut. Key to our method is a theorem showing that any polygonal subdivision can be converted into an m-guillotine subdivision by adding a set of edges whose total length is small: at most
Another planar subdivision, namely the guillotine subdivision, was introduced by Mitchell REF for geometric optimization problems in the plane.
Guillotine Subdivisions Approximate Polygonal Subdivisions: A Simple Polynomial-Time Approximation Scheme for Geometric TSP, k-MST, and Related Problems
SIAM J. Comput.
Mathematics Computer Science