1 research outputs found
Simplification of Polyline Bundles
We propose and study a generalization to the well-known problem of polyline
simplification. Instead of a single polyline, we are given a set of
polylines possibly sharing some line segments and bend points. Our goal is to
minimize the number of bend points in the simplified bundle with respect to
some error tolerance (measuring Fr\'echet distance) but under the
additional constraint that shared parts have to be simplified consistently. We
show that polyline bundle simplification is NP-hard to approximate within a
factor for any where is the
number of bend points in the polyline bundle. This inapproximability even
applies to instances with only polylines. However, we identify the
sensitivity of the solution to the choice of as a reason for this
strong inapproximability. In particular, we prove that if we allow to
be exceeded by a factor of in our solution, we can find a simplified
polyline bundle with no more than bend points in
polytime, providing us with an efficient bi-criteria approximation. As a
further result, we show fixed-parameter tractability in the number of shared
bend points