Streaming Submodular Matching Meets the Primal-Dual Method
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We study streaming submodular maximization subject to matching/b-matching constraints (MSM/MSbM), and present improved upper and lower bounds for these problems. On the upper bounds front, we give primal-dual algorithms achieving the following approximation ratios. ∙ 3+2√(2)≈ 5.828 for monotone MSM, improving the previous best ratio of 7.75. ∙ 4+3√(2)≈ 7.464 for non-monotone MSM, improving the previous best ratio of 9.899. ∙ 3+ϵ for maximum weight b-matching, improving the previous best ratio of 4+ϵ. On the lower bounds front, we improve on the previous best lower bound of e/e-1≈ 1.582 for MSM, and show ETH-based lower bounds of ≈ 1.914 for polytime monotone MSM streaming algorithms. Our most substantial contributions are our algorithmic techniques. We show that the (randomized) primal-dual method, which originated in the study of maximum weight matching (MWM), is also useful in the context of MSM. To our knowledge, this is the first use of primal-dual based analysis for streaming submodular optimization. We also show how to reinterpret previous algorithms for MSM in our framework; hence, we hope our work is a step towards unifying old and new techniques for streaming submodular maximization, and that it paves the way for further new results.
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