We propose and study generalizations 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. The simplification of those shared parts has to be consistent among the polylines. We consider two optimization goals: either minimizing the number of line segments or minimizing the number of bend points in the simplification. By reduction from Minimum-Independent-Dominating-Set, we show that both of these optimization problems are NP-hard to approximate within a factor n^{1/3 - \varepsilon} for any \varepsilon > 0 where n is the number of bend points in the polyline bundle. Moreover, we outline that both problems remain NP-hard even if the input is planar. On the positive side, we give a polynomial-size integer linear program and show fixed-parameter tractability in the number of shared bend points.
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 \ell 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 \delta (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 n^{1/3 - \varepsilon} for any \varepsilon > 0 where n is the number of bend points in the polyline bundle. This inapproximability even applies to instances with only \ell=2 polylines. However, we identify the sensitivity of the solution to the choice of \delta as a reason for this strong inapproximability. In particular, we prove that if we allow \delta to be exceeded by a factor of 2 in our solution, we can find a simplified polyline bundle with no more than O(\log (\ell + n)) \cdot OPT 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.
R
generalizations
a generalization
A
\ell
R
The simplification of those shared parts has to be consistent among the polylines. We consider two optimization goals: either minimizing the number of line segments or minimizing the number of
Our goal is to minimize the number of
R
simplification. By reduction from Minimum-Independent-Dominating-Set, we show that both of these optimization problems are
simplified bundle with respect to some error tolerance \delta (measuring Fr\'echet distance) but under the additional constraint that shared parts have to be simplified consistently. We show that polyline bundle simplification is
R
Moreover, we outline that both problems remain NP-hard even if the input is planar. On the positive side, we give a polynomial-size integer linear program and
This inapproximability even applies to instances with only \ell=2 polylines. However, we identify the sensitivity of the solution to the choice of \delta as a reason for this strong inapproximability. In particular, we prove that if we allow \delta to be exceeded by a factor of 2 in our solution, we can find a simplified polyline bundle with no more than O(\log (\ell + n)) \cdot OPT bend points in polytime, providing us with an efficient bi-criteria approximation. As a further result, we