We study parallelizable algorithms for maximization of a submodular function, not necessarily monotone, with respect to a cardinality constraint k. We improve the best approximation factor achieved by an algorithm that has optimal adaptivity and query complexity, up to logarithmic factors in the size n of the ground set, from 0.039 - \epsilon to 0.193 - \epsilon. We provide two algorithms; the first has approximation ratio 1/6 - \epsilon, adaptivity O( \log n ), and query complexity O( n \log k ), while the second has approximation ratio 0.193 - \epsilon, adaptivity O( \log^2 n ), and query complexity O(n \log k). Heuristic versions of our algorithms are empirically validated to use a low number of adaptive rounds and total queries while obtaining solutions with high objective value in comparison with highly adaptive approximation algorithms .
We study parallelizable algorithms for maximization of a submodular function, not necessarily monotone, with respect to a cardinality constraint k. We improve the best approximation factor achieved by an algorithm that has optimal adaptivity and query complexity, up to logarithmic factors in the size n of the ground set, from 0.039 - \epsilon to 0.193 - \epsilon. We provide two algorithms; the first has approximation ratio 1/6 - \epsilon, adaptivity O( \log n ), and query complexity O( n \log k ), while the second has approximation ratio 0.193 - \epsilon, adaptivity O( \log^2 n ), and query complexity O(n \log k). Heuristic versions of our algorithms are empirically validated to use a low number of adaptive rounds and total queries while obtaining solutions with high objective value in comparison with state-of-the-art approximation algorithms, including continuous algorithms that use the multilinear extension .
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highly adaptive approximation algorithms
state-of-the-art approximation algorithms, including continuous algorithms that use the multilinear extension