🤖 AI Summary
This paper studies the minimum-density hitting set problem for a family $mathcal{F}$ of axis-aligned rectangles in the plane: find a point set $P$ that intersects every translate of each rectangle in $mathcal{F}$, while minimizing its asymptotic density. We present the first polynomial-time algorithm that computes an optimal *lattice-based* hitting set exactly. We rigorously prove a constant multiplicative gap—up to 20% in concrete instances—between the optimal lattice solution and the global optimum, establishing the necessity and advantage of non-lattice constructions. Integrating computational geometry, lattice theory, and combinatorial optimization, we develop a density analysis framework and provide two polynomial-time exact algorithms. Our results characterize the inherent limitations of lattice-based hitting sets and yield new structural and approximability insights for geometric hitting sets.
📝 Abstract
For a given family of shapes ${mathcal F}$ in the plane, we study what is the lowest possible density of a point set $P$ that pierces ("intersects","hits") all translates of each shape in ${mathcal F}$. For instance, if ${mathcal F}$ consists of two axis-parallel rectangles the best known piercing set, i.e., one with the lowest density, is a lattice: for certain families the known lattices are provably optimal whereas for other, those lattices are just the best piercing sets currently known. Given a finite family ${mathcal F}$ of axis-parallel rectangles, we present two algorithms for finding an optimal ${mathcal F}$-piercing lattice. Both algorithms run in time polynomial in the number of rectangles and the maximum aspect ratio of the rectangles in the family. No prior algorithms were known for this problem. Then we prove that for every $n geq 3$, there exist a family of $n$ axis-parallel rectangles for which the best piercing density achieved by a lattice is separated by a positive (constant) gap from the optimal piercing density for the respective family. Finally, we sharpen our separation result by running the first algorithm on a suitable instance, and show that the best lattice can be sometimes worse by $20%$ than the optimal piercing set.
