Streaming Submodular Maximization under Matroid Constraints

Moran Feldman; Paul Liu; Ashkan Norouzi-Fard; Ola Svensson; Rico Zenklusen

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arXiv · 2107.07183

Streaming Submodular Maximization under Matroid Constraints

Moran Feldman·Paul Liu·Ashkan Norouzi-Fard·Ola Svensson·Rico Zenklusen

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Abstract

Recent progress in (semi-)streaming algorithms for monotone submodular function maximization has led to tight results for a simple cardinality constraint. However, current techniques fail to give a similar understanding for natural generalizations, including matroid constraints. This paper aims at closing this gap. For a single matroid of rank $k$ (i.e., any solution has cardinality at most $k$), our main results are: 1) a single-pass streaming algorithm that uses $\widetilde{O}(k)$ memory and achieves an approximation guarantee of $0.3178$, and 2) a multi-pass streaming algorithm that uses $\widetilde{O}(k)$ memory and achieves an approximation guarantee of $(1-1/e - \varepsilon)$ by taking a constant (depending on $\varepsilon$) number of passes over the stream. This improves on the previously best approximation guarantees of $1/4$ and $1/2$ for single-pass and multi-pass streaming algorithms, respectively. In fact, our multi-pass streaming algorithm is tight in that any algorithm with a better guarantee than $1/2$ must make several passes through the stream and any algorithm that beats our guarantee of $1-1/e$ must make linearly many passes (as well as an exponential number of value oracle queries). Moreover, we show how the approach we use for multi-pass streaming can be further strengthened if the elements of the stream arrive in uniformly random order, implying an improved result for $p$-matchoid constraints.

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BibTeX RIS

Moran Feldman, Paul Liu, Ashkan Norouzi-Fard, Ola Svensson, Rico Zenklusen. 2021-07-15. Streaming Submodular Maximization under Matroid Constraints. https://arxiv.org/abs/2107.07183

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