A Constant-Factor Approximation Algorithm for the Geometrick-MST Problem in the Plane

Joseph S. B. Mitchell, Avrim Blum, Prasad Chalasani, Santosh Vempala
1998 SIAM journal on computing (Print)  
We show that any rectilinear polygonal subdivision in the plane can be converted into a \guillotine" subdivision whose length is at most twice that of the original subdivision. \Guillotine" subdivisions have a simple recursive structure that allows one to search for \optimal" such subdivisions in polynomial time, using dynamic programming. In particular, a consequence of our main theorem is a very simple proof that the k-MST problem in the plane has a constant factor polynomial-time
more » ... n algorithm: We obtain a factor of 2 (resp., 3) for the L 1 metric, and a factor of 2 p 2 (resp., 3:266) for the L 2 (Euclidean) metric in the case in which Steiner points are allowed (resp., not allowed). Key words. minimum spanning trees, k-MST, guillotine subdivisions, quota traveling salesman problem, prize-collecting salesman, and bank robber (orienteering) problem, network optimization, computational geometry, dynamic programming, approximation algorithms polynomial AMS subject classi cations. 68Q25, 68R10, 68U05 1. Introduction. We introduce a new technique that can be used to obtain simple approximation algorithms for geometric network design problems. The method is based on the concept of a \guillotine subdivision". Roughly speaking, a \guillotine subdivision" is a rectilinear polygonal subdivision with the property that there exists a horizontal or vertical line (a \cut") whose intersection with the edge set is connected and the subdivisions on either side of the line are also guillotine. The connectedness property allows one to apply dynamic programming to optimize over guillotine subdivisions, as there is a succinct speci cation of how the subdivision interacts with the \cuts" that make up the boundary of a rectangle that speci es a \subproblem" of the dynamic program. Key to our method is a theorem showing that any rectilinear polygonal subdivision can be converted into a guillotine subdivision by adding a set of edges whose total length is small (at most that of the original subdivision). To illustrate the power of the method, we show how it can be used to give a very simple constant-factor approximation algorithm for the geometric k-MST problem, obtaining a substantially better factor than previously known. We also apply it to some related problems (the \quota TSP", \prize-collecting salesman", and \bank robber (orienteering)" problems). A Motivating Application. A special case of the \quota TSP" problem is the following: You are a salesman who must sell k items. You can sell one item in each of n cities (n k). You want to nd a shortest tour that visits at least k cities, so that you can sell your quota of k items. A solution (exact or approximate) to the k-MST
doi:10.1137/s0097539796303299 fatcat:fibg35t3arh73dpft2roqat73q