The Parameterized Complexity of the Rectangle Stabbing Problem and Its Variants
FAW '08 Proceedings of the 2nd annual international workshop on Frontiers in Algorithmics
Parameterized Complexity of Stabbing Rectangles and Squares in the Plane
WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
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We consider a special set covering problem. This problem is a generalization of finding a minimum clique cover in an interval graph. When formulated as an integer program, the 0-1 constraint matrix of this integer program can be partitioned into an interval matrix and a special 0-1 matrix with a single 1 per column. We show that the value of this formulation is bounded by $$\tfrac{{2k}}{{k + 1}}$$ times the value of the LP-relaxation, where k is the maximum row sum of the special matrix. For the "smallest" difficult case, i.e., k = 2, this bound is tight. Also we provide an O(n) 3/2 -approximation algorithm in case k = 2.