Towards a Fixed Parameter Tractability of Geometric Hitting Set Problem for Axis-Parallel Squares Intersecting a Given Straight Line

Abstract

The Hitting Set Problem (HSP) is the well-known extremal problem adopting interest of researchers in the fields of statistical learning theory, combinatorial optimization, and computational geometry for decades. It is known, that the problem is NP-hard in its general case and remains intractable even in very specific geometric settings, e.g., for axis-parallel rectangles intersecting a given straight line. Recently, for the special case, where all the rectangles are unit squares, a polynomial but very time-consuming exact algorithm was proposed. We improve this algorithm to decrease its complexity bound more than 100 degrees of magnitude. Also, we extend it on the more general case and prove that the geometric HSP for axis-parallel (not necessarily unit) squares intersected by a line is polynomially solvable for any fixed range of the squares to hit. Hence, this geometric setting of the HSP belongs to the class of Fixed Parameter Tractable (FPT) problems.