A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
Improved Approximation Algorithms for Geometric Set Cover
Discrete & Computational Geometry
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
On the set multi-cover problem in geometric settings
Proceedings of the twenty-fifth annual symposium on Computational geometry
Weighted geometric set cover via quasi-uniform sampling
Proceedings of the forty-second ACM symposium on Theory of computing
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling
Proceedings of the twenty-third 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
A constant-factor approximation for multi-covering with disks
Proceedings of the twenty-ninth annual symposium on Computational geometry
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We give a randomized polynomial time algorithm with approximation ratio O(logφ(n)) for weighted set multi-cover instances with a shallow cell complexity of at most f(n,k)=nφ(n) kO(1). Up to constant factors, this matches a recent result of Könemann et al. for the set cover case, i.e. when all the covering requirements are 1. One consequence of this is an O(1)-approximation for geometric weighted set multi-cover problems when the geometric objects have linear union complexity; for example when the objects are disks, unit cubes or halfspaces in ℝ3. Another consequence is to show that the real difficulty of many natural capacitated set covering problems lies with solving the associated priority cover problem only, and not with the associated multi-cover problem.