Generalized class cover problem with axis-parallel strips

We initiate the study of a generalization of the class cover problem [Cannon and Cowen [1], Bereg et al. [2]] the generalized class cover problem, where we are allowed to misclassify some points provided we pay an associated positive penalty for every misclassified point. Two versions: single coverage and multiple coverage, of the generalized class cover problem are investigated. We study five different variants of both versions of the generalized class cover problem with axis-parallel strips and axis-parallel half-strips extending to different directions in the plane, thus extending similar work by Bereg et al. (2012) [2] on the class cover problem. We prove that the multiple coverage version of the generalize class cover problem with axis-parallel strips are in $P$, whereas the single coverage version is $\mathrm{NP}$-hard. A factor 2 approximation algorithm is provided for the later problem. The $\mathrm{APX}$-hardness result is also shown for the single coverage version. For half-strips extending to exactly one direction, both the single and multiple coverage versions can be solved in polynomial time using dynamic programming. In the case of half-strips extending to two orthogonal directions, we prove the class cover problem is $\mathrm{NP}$-hard followed by $\mathrm{APX}$-hard. This gives improve hardness results compare to Bereg et al. (2012) [2], where they proved the class cover problem with half-strips oriented in four different directions is $\mathrm{NP}$-hard. These $\mathrm{NP}$- and $\mathrm{APX}$-hardness results can directly apply to both single and multiple versions. Finally, constant factor approximation algorithms are provided for half-strips extending to more than one direction.