Fast approximation algorithms for a nonconvex covering problem
Journal of Algorithms
Some APX-completeness results for cubic graphs
Theoretical Computer Science
Lenses in arrangements of pseudo-circles and their applications
Journal of the ACM (JACM)
Improved Approximation Algorithms for Geometric Set Cover
Discrete & Computational Geometry
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Approximation algorithms for maximum independent set of pseudo-disks
Proceedings of the twenty-fifth annual symposium on Computational geometry
Hitting sets when the VC-dimension is small
Information Processing Letters
Weighted geometric set cover via quasi-uniform sampling
Proceedings of the forty-second ACM symposium on Theory of computing
Improved Results on Geometric Hitting Set Problems
Discrete & Computational Geometry
PTAS for weighted set cover on unit squares
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes
SIAM Journal on Computing
Tight lower bounds for the size of epsilon-nets
Proceedings of the twenty-seventh annual symposium on Computational geometry
Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
On column-restricted and priority covering integer programs
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
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We study several geometric set cover and set packing problems involving configurations of points and geometric objects in Euclidean space. We show that it is APX-hard to compute a minimum cover of a set of points in the plane by a family of axis-aligned fat rectangles, even when each rectangle is an @e-perturbed copy of a single unit square. We extend this result to several other classes of objects including almost-circular ellipses, axis-aligned slabs, downward shadows of line segments, downward shadows of graphs of cubic functions, fat semi-infinite wedges, 3-dimensional unit balls, and axis-aligned cubes, as well as some related hitting set problems. We also prove the APX-hardness of a related family of discrete set packing problems. Our hardness results are all proven by encoding a highly structured minimum vertex cover problem which we believe may be of independent interest. In contrast, we give a polynomial-time dynamic programming algorithm for geometric set cover where the objects are pseudodisks containing the origin or are downward shadows of pairwise 2-intersecting x-monotone curves. Our algorithm extends to the weighted case where a minimum-cost cover is required. We give similar algorithms for several related hitting set and discrete packing problems.