On the Minimum Consistent Subset Problem

Abstract

Let P be a set of n colored points in the plane. Introduced by Hart [7], a consistent subset of P, is a set $S\subseteq P$ such that for every point p in $P\setminus S$, 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 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 $O(n\log n)$-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 $O(n \log n)$-time algorithm for computing a variant of the compact Voronoi diagram; this improves the previously claimed expected running time.

• 3. An $O(n\log ^2 n)$-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 that are given from left to right; 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 paraboloid lifting, planar separators, additively-weighted Voronoi diagrams with respect to convex distance functions, point location in farthest-point Voronoi diagrams, range trees, minimum covering of a circle with arcs, and several geometric transformations.