Hitting Set Problem for Axis-Parallel Squares Intersecting a Straight Line Is Polynomially Solvable for Any Fixed Range of Square Sizes

Abstract

The Hitting Set Problem is the well known discrete optimization problem adopting interest of numerous scholars in graph theory, computational geometry, operations research, and machine learning. The problem is NP-hard and remains intractable even in very specific settings, e.g., for axis-parallel rectangles on the plane. Recently, for unit squares intersecting a straight line, a polynomial time optimal algorithm was proposed. Unfortunately, the time consumption of this algorithm was $O(n^{145})$. We propose an improved algorithm, whose complexity bound is more than 100 orders of magnitude less. We extend this algorithm to the more general case of the problem and show that the geometric HSP for axis-parallel (not necessarily unit) squares intersected by a line is polynomially solvable for any fixed range of squares to hit. We believe that the obtained theoretical complexity bounds for our algorithms still can be improved further. According to the results of the numerical evaluation presented in the concluding section of the paper, at least for unit squares, an average time consumption bound of our algorithm is less then its deterministic counterpart by 9 orders of magnitude.