TY - GEN
T1 - Parameterized complexity of red blue set cover for lines
AU - Ashok, Pradeesha
AU - Kolay, Sudeshna
AU - Saurabh, Saket
N1 - Funding Information:
S. Saurabh—The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 306992.
Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2016.
PY - 2016/1/1
Y1 - 2016/1/1
N2 - We investigate the parameterized complexity of Generalized Red Blue Set Cover (Gen-RBSC), a generalization of the classic SetCover problem and themore recently studied Red Blue Set Cover problem. Given a universe U containing b blue elements and r red elements, positive integers kℓ and kr, and a family F of ℓ sets over U, the Gen-RBSC problem is to decide whether there is a subfamily Fℓ⊆ F of size at most kℓ that covers all blue elements, but at most kr of the red elements. This generalizes Set Cover and thus in full generality it is intractable in the parameterized setting, when parameterized by kℓ + kr. In this paper, we study Gen-RBSC-lines, where the elements are points in the plane and sets are defined by lines.We study this problem for the parameters kℓ, kr, and kℓ + kr. For all these cases, we either prove that the problem is Whard or show that the problem is fixed parameter tractable (FPT). Finally, for the parameter kℓ + kr, for which Gen-RBSC-lines admits FPT algorithms, we show that the problem does not have a polynomial kernel unless co-NP ⊆ NP/poly. Further,we showthat theFPTalgorithm does not generalize to higher dimensions.
AB - We investigate the parameterized complexity of Generalized Red Blue Set Cover (Gen-RBSC), a generalization of the classic SetCover problem and themore recently studied Red Blue Set Cover problem. Given a universe U containing b blue elements and r red elements, positive integers kℓ and kr, and a family F of ℓ sets over U, the Gen-RBSC problem is to decide whether there is a subfamily Fℓ⊆ F of size at most kℓ that covers all blue elements, but at most kr of the red elements. This generalizes Set Cover and thus in full generality it is intractable in the parameterized setting, when parameterized by kℓ + kr. In this paper, we study Gen-RBSC-lines, where the elements are points in the plane and sets are defined by lines.We study this problem for the parameters kℓ, kr, and kℓ + kr. For all these cases, we either prove that the problem is Whard or show that the problem is fixed parameter tractable (FPT). Finally, for the parameter kℓ + kr, for which Gen-RBSC-lines admits FPT algorithms, we show that the problem does not have a polynomial kernel unless co-NP ⊆ NP/poly. Further,we showthat theFPTalgorithm does not generalize to higher dimensions.
UR - http://www.scopus.com/inward/record.url?scp=84961718094&partnerID=8YFLogxK
U2 - 10.1007/978-3-662-49529-2_8
DO - 10.1007/978-3-662-49529-2_8
M3 - Conference contribution
AN - SCOPUS:84961718094
SN - 9783662495285
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 96
EP - 109
BT - LATIN 2016
A2 - Navarro, Gonzalo
A2 - Kranakis, Evangelos
A2 - Chávez, Edgar
PB - Springer Verlag
T2 - 12th Latin American Symposium on Theoretical Informatics, LATIN 2016
Y2 - 11 April 2016 through 15 April 2016
ER -