New algorithms for two-label point labeling

Abstract

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 I to i. For circles the improvement from | to ~ 0.513 is less significant, but the fact that | 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 0(n log n)timeand O(n) space.