New algorithms for two-label point labeling

Given a label shape L and a set of n points in the plane, the 2-label point-labeling problem consists of placing 2n non-intersecting translated copies of L of maximum size such that each point touches two unique copies—its labels. In this paper we give new and simple approximation algorithms for L an axis-parallel square or a circle. For squares we improve the best previously known approximation factor from 1/3 to 1/2. For circles the improvement from 1/2 to ˜ 0.513 is less significant, but the fact that 1/2 is not best possible is interesting in its own right. For the decision version of the latter problem we have an NP-hardness proof that also shows that it is NP-hard to approximate the label size beyond a factor of ˜ 0.732. As their predecessors, our algorithms take O(n log n) time and O(n) space