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 + e)-approximation and a (2.98 + e)-approximation; for MLUCP, a (1.5 + e)-approximation and a (1.491 + e)-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.

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