The Hitting Set Problem (HSP) is the well known extremal problem adopting research interest in the fields of combinatorial optimization, computational geometry, and statistical learning theory for decades. In the general setting, the problem is NP-hard and hardly approximable. Also, the HSP remains intractable even in very specific geometric settings, e.g. for axis-parallel rectangles intersecting a given straight line. Recently, for the special case of the problem, where all the rectangles are unit squares, a polynomial but very time consuming optimal algorithm was proposed. We improve this algorithm to decrease its complexity bound more than 100 degrees of magnitude. Also, we extend it 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.Key words: Hitting set problem, Dynamic programming, Computational geometry, Parameterized complexity.
IntroductionWe consider the parameterized complexity of a geometric statement of the well-known Hitting Set Problem (HSP), engaging researchers in combinatorial optimization, computational geometry and statistical learning from early 1980-th.To the best of our knowledge, HSP gains theoretical interest because it was the first intractable combinatorial optimization problem, whose approximation algorithms were dramatically improved On the other hand, the concepts of hitting set and classifier ensemble, making decisions by some voting logic, seem to be related very closely. Consequently, approximation techniques developed for HSP and its dual Set Cover problem are closely related to the well-known boosting learning technique [14], especially in the context of the minimal committee problem looking for minimum VC-dimension correct majoritary classifier ensemble (see, e.g., [8][9][10]).In addition, new efficient optimal and approximation algorithms for Hitting Set and Set Cover problems have a practical importance, e.g. in design of reliable wireless networks [13].The Hitting Set Problem for Axis-Parallel Rectangles (HSP-APR) is a well-studied geometric setting of the HSP. This setting is also NP-hard [5] and remains intractable even for unit squares.
Daniel M. Khachay and Michael Yu. KhachayIn papers [2,7], first polynomial time approximation schemes (PTAS) are proposed for axis-parallel squares. Paper [3] introduces 6-approximation polynomial time algorithm for the case of rectangles intersecting some axis-monotone curve. In [4], this particular case of HSP-APR is proved to be NP-hard even for a straight line and the first 4-approximation algorithm is constructed.In this paper, we improve one of the recent results describing a polynomial time solvable subclass of this problem. Recently, Mudgal and Pandit [12] introduced an optimal polynomial time algorithm for the Hitting Set Problem for Axis Parallel Unit Squares Intersecting a given Straight Line (HSP-APUS-ISL). The theoretical importance of this result can hardly be...