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ResourceConstrained Geometric Network Optimization (Extended abstract)

Summary: Resource­Constrained Geometric Network Optimization
(Extended abstract)
Esther M. Arkin \Lambda Joseph S. B. Mitchell y Giri Narasimhan z
December 4, 1997
We study a variety of geometric network optimization problems on a set of points, in which we are
given a resource bound, B, on the total length of the network, and our objective is to maximize the number
of points visited (or the total ``value'' of points visited). In particular, we resolve the well­publicized open
problem on the approximability of the rooted ``orienteering problem'' for the case in which the sites are
given as points in the plane and the network required is a cycle. We obtain a 2­approximation for this
problem. We also obtain approximation algorithms for variants of this problem in which the network
required is a tree (3­approximation) or a path (2­approximation). No prior approximation bounds were
known for any of these problems.
We also obtain improved approximation algorithms for geometric instances of the unrooted orienteer­
ing problem, where we obtain a 2­approximation for both the cycle and tree versions of the problem on
points in the plane, as well as a 5­approximation for the tree version in edge­weighted graphs. Further,
we study generalizations of the basic orienteering problem, to the case of multiple roots, sites that are
polygonal regions, etc., where we again give the first known approximation results.
Our methods are based on some new tools which may be of interest in their own right: (1) some new
results on m­guillotine subdivisions in the plane, strengthening the original approximation bound, and


Source: Arkin, Estie - Department of Applied Mathematics and Statistics, SUNY at Stony Brook


Collections: Mathematics