Localization is a fundamental problem in robotics. The kidnapped robot possesses a compass and map of its environment it must determine its location at a minimum cost of travel distance. The problem is NP-hard even to minimize within factor $c\\log n$, where $n$ is the map size. No efficient approximation algorithm is known.
We give an $O(\\log3 n)$-factor algorithm which runs in polynomial time. The key idea is to plan travel in a ``majority-rule'' map, which eliminates uncertainty and permits a link to the $\\frac{1}{2}$-Group Steiner (not Group Steiner) problem. The approximation factor is not far from optimal: we prove a $c\\log^{2-\\epsilon} n$ lower bound, assuming $NP \\not\\subseteq ZTIME(n^{polylog(n)})$, for the grid graphs commonly used in practice. We also extend the algorithm to polygonal maps by discretizing the problem using novel geometric techniques.
