Map labeling with circles
We study two geometric optimization problems motivated by cartographic applications: Map Labeling with Uniform Circles (MLUC) and Map Labeling with Uniform Circle Pairs (MLUCP). We show that the decision problems of both MLUC and MLUCP are NP-hard, and that the related optimization problems for maximizing the label sizes are NP-hard to approximate within factor 1.0349. We design approximation algorithms with constant performance guarantees for the two problems: for MLUC, we present a (3 + o)-approximation and a (2.98 + o)-approximation; for MLUCP, a (1.5+o)-approximation and a (1.491+o)-approximation. We also describe the implementation of AMLUC, a software system for automated map labeling with uniform circles. The system is based on our approximation algorithms for MLUC and uses an effective shake-and-grow heuristic to find near-optimal label placements.