On the Minimum Consistent Subset Problem

Let P be a set of n colored points in the d-dimensional Euclidean space. Introduced by Hart (1968), consistent subset P, is $$S\subseteq P$$ such that for every point p $$P {\setminus } S$$ , closest S has same color as p. The problem to find with minimum cardinality. This known NP-complete even two-colored sets. Since initial presentation this problem, aside from hardness results, there not been significant progress algorithmic view. In paper we present following results plane: (1) first subexponential-time algorithm problem. (2) An $$O(n\log n)$$ -time finds size two sets (if exists). Along way prove result which an independent interest: given translations cone (defined intersection halfspaces) and $$\mathbb {R}^3$$ time one can decide whether or cone. (3) ^2 where class contains exactly point; improves previous best $$O(n^2)$$ running due Wilfong (SoCG 1991). (4) O(n)-time on collinear are left right; time. (5) A non-trivial $$O(n^6)$$ dynamic programming arranged parallel lines. To obtain these combine tools planar separators, paraboloid lifting, additively-weighted Voronoi diagrams respect convex distance functions, location farthest-point diagrams, range trees, covering circle arcs, several geometric transformations.