ON PARAMETERIZED COMPLEXITY OF HITTING SET PROBLEM FOR AXIS–PARALLEL SQUARES INSTERSECTING A STRAIGHT LINE

Daniel M Khachay,Michael Yu Khachay

doi:10.15826/umj.2016.2.010

Daniel M Khachay, Michael Yu Khachay

Open Access

https://doi.org/10.15826/umj.2016.2.010

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Abstract

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.

Highlights

We 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 [11] on the basis of Vapnik and Chervonenkis’s [15] results in statistical learning theory
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

Summary

Introduction

We 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. The Hitting Set Problem for Axis-Parallel Rectangles (HSP–APR) is a well-studied geometric setting of the HSP This setting is NP-hard [5] and remains intractable even for unit squares. 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 overestimated, since almost all known geometric settings of the HSP, including extremely specific ones, are intractable. This algorithm is impractical due to its incredibly high time consumption of O(n145).

Problem statement

Improved algorithm for unit squares

General case of HSP–APS–ISL

Conclusion