Maximizing a monotone submodular function under various constraints is a classical and intensively studied problem. However, in the single-pass streaming model, where the elements arrive one by one and an algorithm can store only a small fraction of input elements, there is large gap in our knowledge, even though several approximation algorithms have been proposed in the literature. In this work, we present the first lower bound on the approximation ratios for cardinality and matroid constraints that beat 1− 1e in the single-pass streaming model. Let n be the number of elements in the stream. Then, we prove that any (randomized) streaming algorithm for a cardinality constraint with approximation ratio 2−√2+ε requires Ω(Kn2 ) space for any ε > 0, where K is the size limit of the output set. We also prove that any (randomized) streaming algorithm for a (partition) matroid constraint with approximation ratio 2KK−1 + ε requires Ω(Kn2 ) space for any ε > 0, where K is the rank of the given matroid. In addition, we give streaming algorithms that assume access to the objective function via a weak oracle that can only be used to evaluate function values on feasible sets. Specifically, we show weak-oracle streaming algorithms for cardinality and matroid constraints with approximation ratios 2KK−1 and 21, respectively, whose space complexity is exponential in K but is independent of n. The former one exactly matches the known inapproximability result for a cardinality constraint in the weak oracle model. The latter one almost matches our lower bound of 2KK−1 for a matroid constraint, which almost settles the approximation ratio for a matroid constraint that can be obtained by a streaming algorithm whose space complexity is independent of n.
ASJC Scopus subject areas
- 数学 (全般)