Approximating geometric coverage problems
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Helly-type theorems for approximate covering
Proceedings of the twenty-fourth annual symposium on Computational geometry
Self-improving algorithms for delaunay triangulations
Proceedings of the twenty-fourth annual symposium on Computational geometry
Proceedings of the twenty-fourth annual symposium on Computational geometry
Stabbing Convex Polygons with a Segment or a Polygon
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Small-size ε-nets for axis-parallel rectangles and boxes
Proceedings of the forty-first annual ACM symposium on Theory of computing
Epsilon nets and union complexity
Proceedings of the twenty-fifth annual symposium on Computational geometry
PTAS for geometric hitting set problems via local search
Proceedings of the twenty-fifth annual symposium on Computational geometry
Near-linear approximation algorithms for geometric hitting sets
Proceedings of the twenty-fifth annual symposium on Computational geometry
Approximation algorithms for maximum independent set of pseudo-disks
Proceedings of the twenty-fifth annual symposium on Computational geometry
On the set multi-cover problem in geometric settings
Proceedings of the twenty-fifth annual symposium on Computational geometry
Practical Discrete Unit Disk Cover Using an Exact Line-Separable Algorithm
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Note: Approximation algorithms for art gallery problems in polygons
Discrete Applied Mathematics
Domination in geometric intersection graphs
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Weighted geometric set cover via quasi-uniform sampling
Proceedings of the forty-second ACM symposium on Theory of computing
A note about weak ε-nets for axis-parallel boxes in d-space
Information Processing Letters
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
Small Approximate Pareto Sets for Biobjective Shortest Paths and Other Problems
SIAM Journal on Computing
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
Hitting sets online and vertex ranking
ESA'11 Proceedings of the 19th European conference on Algorithms
SIAM Journal on Computing
SIAM Journal on Computing
Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Improved bound for the union of fat triangles
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
The class cover problem with boxes
Computational Geometry: Theory and Applications
Approximation algorithms for art gallery problems in polygons and terrains
WALCOM'10 Proceedings of the 4th international conference on Algorithms and Computation
Proceedings of the thirteenth ACM international symposium on Mobile Ad Hoc Networking and Computing
Proceedings of the twenty-eighth annual symposium on Computational geometry
On the set multicover problem in geometric settings
ACM Transactions on Algorithms (TALG)
Weighted geometric set multi-cover via quasi-uniform sampling
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Small-size relative (p,ε)-approximations for well-behaved range spaces
Proceedings of the twenty-ninth annual symposium on Computational geometry
Exact algorithms and APX-hardness results for geometric packing and covering problems
Computational Geometry: Theory and Applications
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Given a collection S of subsets of some set ${\Bbb U},$ and ${\Bbb M}\subset{\Bbb U},$ the set cover problem is to find the smallest subcollection $C\subset S$ that covers ${\Bbb M},$ that is, ${\Bbb M} \subseteq \bigcup (C),$ where $\bigcup(C)$ denotes $\bigcup_{Y \in C} Y.$ We assume of course that S covers ${\Bbb M}.$ While the general problem is NP-hard to solve, even approximately, here we consider some geometric special cases, where usually ${\Bbb U} = {\Bbb R}^d.$ Combining previously known techniques [4], [5], we show that polynomial-time approximation algorithms with provable performance exist, under a certain general condition: that for a random subset $R\subset S$ and nondecreasing function f(·), there is a decomposition of the complement ${\Bbb U}\backslash\bigcup (R)$ into an expected at most f(|R|) regions, each region of a particular simple form. Under this condition, a cover of size O(f(|C|)) can be found in polynomial time. Using this result, and combinatorial geometry results implying bounding functions f(c) that are nearly linear, we obtain o(log c) approximation algorithms for covering by fat triangles, by pseudo-disks, by a family of fat objects, and others. Similarly, constant-factor approximations follow for similar-sized fat triangles and fat objects, and for fat wedges. With more work, we obtain constant-factor approximation algorithms for covering by unit cubes in ${\Bbb R}^3,$ and for guarding an x-monotone polygonal chain.