Simplification of Polyline Bundles

Abstract

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.