Covering Points with Rectangular Boundaries

Abstract

Geometric covering problems typically ask for a small family of geometric objects whose union contains all input points. In this paper we study a more rigid variant, boundary covering, where every point must lie on the boundary of at least one chosen object. Motivated by the framework of Langerman and Morin [Discret. Comput. Geom., 2005] for boundary covering by hyperspheres, we initiate a systematic study of boundary covering by axis-parallel rectangles in the plane.
We first consider the discrete setting, where the rectangles must be chosen from a given family. We define Boundary Covering with Discrete Axis-Parallel Rectangles (BCDAPR) as follows: given a point set P ⊆ ℝ², a collection ℛ of axis-parallel rectangles, and an integer k, decide whether P can be covered by the boundaries of at most k rectangles from ℛ. We prove that this discrete boundary-covering problem is W[1]-hard when parameterized by k. 
This motivates the continuous variant, where we are allowed to place rectangles freely. We define Boundary Covering with Continuous Axis-Parallel Rectangles (BCCAPR) as follows: given a point set P ⊆ ℝ² and an integer k, decide whether P can be covered by the boundaries of at most k axis-parallel rectangles. In contrast to the discrete case, we show that BCCAPR is fixed-parameter tractable parameterized by k, with running time 2^𝒪(k log k) ⋅ n^𝒪(1), where n = |P|. Our results does a fine-grained structural analysis of how k rectangles can interact with the point set. On the hardness side, we show that moving from lines to slightly richer shapes already incurs intractability: we prove NP-completeness for boundary covering by axis-aligned L-shapes, and then lift it to NP-completeness of BCCAPR. For the algorithm we reduce BCCAPR to at most 2^𝒪(k log k) instances of Distinct Domain Monotone ,$-CSP, each solvable in polynomial time.