Optimal algorithms for approximate clustering
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
The hardness of approximation: gap location
Computational Complexity
Implementing data cubes efficiently
SIGMOD '96 Proceedings of the 1996 ACM SIGMOD international conference on Management of data
On linear-time deterministic algorithms for optimization problems in fixed dimension
Journal of Algorithms
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
Maximizing the spread of influence through a social network
Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining
Learning diverse rankings with multi-armed bandits
Proceedings of the 25th international conference on Machine learning
Predicting diverse subsets using structural SVMs
Proceedings of the 25th international conference on Machine learning
Optimizing sensing: theory and applications
Optimizing sensing: theory and applications
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
Parameterized Complexity and Approximation Algorithms
The Computer Journal
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We study the complexity of the maximum coverage problem, restricted to set systems of bounded VC-dimension. Our main result is a fixed-parameter tractable approximation scheme: an algorithm that outputs a (1-ε)-approximation to the maximum-cardinality union of k sets, in running time $O(f(ε,k,d)⋅ poly(n)) where n is the problem size, d is the VC-dimension of the set system, and f(ε,k,d) is exponential in (kd/ε)c for some constant c. We complement this positive result by showing that the function f(ε,k,d) in the running-time bound cannot be replaced by a function depending only on (ε,d) or on (k,d), under standard complexity assumptions. We also present an improved upper bound on the approximation ratio of the greedy algorithm in special cases of the problem, including when the sets have bounded cardinality and when they are two-dimensional halfspaces. Complementing these positive results, we show that when the sets are four-dimensional halfspaces neither the greedy algorithm nor local search is capable of improving the worst-case approximation ratio of 1-1/e that the greedy algorithm achieves on arbitrary instances of maximum coverage.