# On the Minimum Consistent Subset Problem

Let P be a set of n colored points in the plane. Introduced by Hart (1968), a consistent subset of P, is a set S⊆ P such that for every point p in P∖ S, the closest point of p in S has the same color as p. The consistent subset problem is to find a consistent subset of P with minimum cardinality. This problem is known to be NP-complete even for two-colored point sets. Since the initial presentation of this problem, aside from the hardness results, there has not been a significant progress from the algorithmic point of view. In this paper we present the following algorithmic results: 1. The first subexponential-time algorithm for the consistent subset problem. 2. An O(n n)-time algorithm that finds a consistent subset of size two in two-colored point sets (if such a subset exists). Towards our proof of this running time we present a deterministic O(n n)-time algorithm for computing a variant of the compact Voronoi diagram; this improves the previously claimed expected running time. 3. An O(n^2 n)-time algorithm that finds a minimum consistent subset in two-colored point sets where one color class contains exactly one point; this improves the previous best known O(n^2) running time which is due to Wilfong (SoCG 1991). 4. An O(n)-time algorithm for the consistent subset problem on collinear points; this improves the previous best known O(n^2) running time. 5. A non-trivial O(n^6)-time dynamic programming algorithm for the consistent subset problem on points arranged on two parallel lines. To obtain these results, we combine tools from planar separators, additively-weighted Voronoi diagrams with respect to convex distance functions, point location in farthest-point Voronoi diagrams, range trees, paraboloid lifting, minimum covering of a circle with arcs, and several geometric transformations.

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