Minimum Cell Connection in Line Segment Arrangements

Abstract

We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points [Formula: see text] and [Formula: see text] in different cells of the induced arrangement: [(i)] compute the minimum number of segments one needs to remove so that there is a path connecting [Formula: see text] to [Formula: see text] that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting [Formula: see text] to [Formula: see text] must stay inside a given polygon [Formula: see text] with a constant number of holes, the segments are contained in [Formula: see text], and the endpoints of the segments are on the boundary of [Formula: see text]. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution.