On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
Discrete & Computational Geometry
Applications of random sampling in computational geometry, II
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
Randomized algorithms
Chernoff-Hoeffding Bounds for Applications with Limited Independence
SIAM Journal on Discrete Mathematics
Approximation algorithms for NP-hard problems
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
Improved Approximation Guarantees for Packing and Covering Integer Programs
SIAM Journal on Computing
Approximation algorithms for covering/packing integer programs
Journal of Computer and System Sciences
Improved Approximation Algorithms for Geometric Set Cover
Discrete & Computational Geometry
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Epsilon nets and union complexity
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
Algorithms for dominating set in disk graphs: breaking the log n Barrier
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
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
On capacitated set cover problems
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Improved bound for the union of fat triangles
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
On column-restricted and priority covering integer programs
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Proceedings of the twenty-eighth annual symposium on Computational geometry
Weighted geometric set multi-cover via quasi-uniform sampling
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
A constant-factor approximation for multi-covering with disks
Proceedings of the twenty-ninth annual symposium on Computational geometry
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The minimum-weight set cover problem is widely known to be O(log n)-approximable, with no improvement possible in the general case. We take the approach of exploiting problem structure to achieve better results, by providing a geometry-inspired algorithm whose approximation guarantee depends solely on an instance-specific combinatorial property known as shallow cell complexity (SCC). Roughly speaking, a set cover instance has low SCC if any column-induced submatrix of the corresponding element-set incidence matrix has few distinct rows. By adapting and improving Varadarajan's recent quasi-uniform random sampling method for weighted geometric covering problems, we obtain strong approximation algorithms for a structurally rich class of weighted covering problems with low SCC. We also show how to derandomize our algorithm. Our main result has several immediate consequences. Among them, we settle an open question of Chakrabarty et al. [8] by showing that weighted instances of the capacitated covering problem with underlying network structure have O(1)-approximations. Additionally, our improvements to Varadarajan's sampling framework yield several new results for weighted geometric set cover, hitting set, and dominating set problems. In particular, for weighted covering problems exhibiting linear (or near-linear) union complexity, we obtain approximability results agreeing with those known for the unweighted case. For example, we obtain a constant approximation for the weighted disk cover problem, improving upon the 2O(log* n)-approximation known prior to our work and matching the O(1)-approximation known for the unweighted variant.