Matching Points with Things

Abstract

Abstract. Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a non-crossing matching between point-object pairs. We show that when the objects we match the points to are finite point sets, the problem is NP-complete in general, and polynomial when the objects are on a line or when their number is at most 2. When the objects are line segments, we show that the problem is NP-complete in general, and polynomial when the segments form a convex polygon or are all on a line. Finally, for objects that are straight lines, we show that the problem of finding a min-max non-crossing matching is NP-complete.

Rainbow polygons for colored point sets in the plane

“…There are few results on covering points with noncrossing segments. It is known, for example, that it is NP-hard to find a maximum noncrossing matching in certain geometric graph [3]. The problem of, given an even number of points, finding a noncrossing matching that minimizes the length of the longest edge is also known to be NP-hard [2].…”