Approximation algorithms for hitting objects with straight lines
Discrete Applied Mathematics
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Discrete & Computational Geometry
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series)
Approximation Algorithms for Rectangle Stabbing and Interval Stabbing Problems
SIAM Journal on Discrete Mathematics
Constant Approximation Algorithms for Rectangle Stabbing and Related Problems
Theory of Computing Systems
Algorithms for capacitated rectangle stabbing and lot sizing with joint set-up costs
ACM Transactions on Algorithms (TALG)
The Parameterized Complexity of the Rectangle Stabbing Problem and Its Variants
FAW '08 Proceedings of the 2nd annual international workshop on Frontiers in Algorithmics
Approximation of a Geometric Set Covering Problem
ISAAC '01 Proceedings of the 12th International Symposium on Algorithms and Computation
Parameterized complexity of independence and domination on geometric graphs
IWPEC'06 Proceedings of the Second international conference on Parameterized and Exact Computation
Parameterized Complexity
Parameterized Complexity of Geometric Problems
The Computer Journal
On the parameterized complexity of some optimization problems related to multiple-interval graphs
Theoretical Computer Science
Fixed-Parameter algorithms for cochromatic number and disjoint rectangle stabbing
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
Information and Computation
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The NP-complete geometric covering problem Rectangle Stabbing is defined as follows: Given a set of horizontal and vertical lines in the plane, a set of rectangles in the plane, and a positive integer k , select at most k of the lines such that every rectangle is intersected by at least one of the selected lines. While it is known that the problem can be approximated in polynomial time with a factor of two, its parameterized complexity with respect to the parameter k was open so far--only its generalization to three or more dimensions was known to be W[1]-hard. Giving two fixed-parameter reductions, one from the W[1]-complete problem Multicolored Clique and one to the W[1]-complete problem Short Turing Machine Acceptance , we prove that Rectangle Stabbing is W[1]-complete with respect to the parameter k , which in particular means that there is no hope for fixed-parameter tractability with respect to the parameter k . Our reductions show also the W[1]-completeness of the more general problem Set Cover on instances that "almost have the consecutive-ones property", that is, on instances whose matrix representation has at most two blocks of 1s per row. For the special case of Rectangle Stabbing where all rectangles are squares of the same size we can also show W[1]-hardness, while the parameterized complexity of the special case where the input consists of rectangles that do not overlap is open. By giving an algorithm running in (4k + 1) k ·n O (1) time, we show that Rectangle Stabbing is fixed-parameter tractable in the still NP-hard case where both these restrictions apply.