[Paper Review] Optimal Streaming Algorithms for Submodular Maximization with Cardinality Constraints

Naor Alaluf, Alina Ene+3 more|arXiv (Cornell University)|Nov 29, 2019

Complexity and Algorithms in Graphs40 references6 citations

TL;DR

This paper presents a single-pass streaming algorithm for non-monotone submodular maximization under a cardinality constraint, using O(k/ε²) memory. It maintains a candidate set via a threshold-based greedy process and achieves an approximation guarantee of α/(1+α) − ε by post-processing with any offline algorithm achieving α-approximation, yielding a 1/2 − ε approximation with exact post-processing and 0.2779-approximation with the state-of-the-art polynomial-time algorithm (α = 0.385).

ABSTRACT

We study the problem of maximizing a non-monotone submodular function subject to a cardinality constraint in the streaming model. Our main contributions are two single-pass (semi-)streaming algorithms that use Õ(k)⋅poly(1/ε) memory, where k is the size constraint. At the end of the stream, both our algorithms post-process their data structures using any offline algorithm for submodular maximization, and obtain a solution whose approximation guarantee is α/(1+α)-ε, where α is the approximation of the offline algorithm. If we use an exact (exponential time) post-processing algorithm, this leads to 1/2-ε approximation (which is nearly optimal). If we post-process with the algorithm of [Niv Buchbinder and Moran Feldman, 2019], that achieves the state-of-the-art offline approximation guarantee of α = 0.385, we obtain 0.2779-approximation in polynomial time, improving over the previously best polynomial-time approximation of 0.1715 due to [Feldman et al., 2018]. One of our algorithms is combinatorial and enjoys fast update and overall running times. Our other algorithm is based on the multilinear extension, enjoys an improved space complexity, and can be made deterministic in some settings of interest.

Motivation & Objective

• Address the lack of optimal streaming algorithms for non-monotone submodular maximization under cardinality constraints.
• Overcome the limitations of prior streaming algorithms that achieve only 1/3 + 2/√2 ≈ 0.1715 approximation in polynomial time.
• Design a streaming algorithm that enables the use of any existing offline submodular maximization algorithm for post-processing to improve approximation guarantees.
• Achieve nearly optimal approximation ratios (1/2 − ε) with polynomial space and time, matching known inapproximability bounds.
• Provide a deterministic, combinatorial algorithm with fast update time per element, rare in non-monotone submodular optimization.

Proposed method

• Use a single-pass streaming approach that maintains a candidate set S1,1 using a threshold-based greedy selection process with a dynamic threshold κ.
• Apply a randomized sampling strategy: create m = Θ(1/ε) independent samples of the stream, each processed via a threshold greedy algorithm to form sets Si,1.
• Construct a union set U = ∪i Si,1, which is used as input for post-processing with any offline submodular maximization algorithm.
• Post-process U using an offline algorithm to obtain a solution T, and return the better of S1,1 and T.
• Leverage convex extensions and concentration bounds to analyze the expected value of the final solution, using the multilinear extension and restricted scale invariance.
• Use conditional probability and submodularity to bound the marginal gain of O′₂ on S1,1 and to show that f(S1,1 ∪ (O1 ∩ U)) is close to f(OPT) in expectation.

Experimental results

Research questions

• RQ1 Can a streaming algorithm achieve a (1/2 − ε)-approximation for non-monotone submodular maximization with cardinality constraints using only poly(k, 1/ε) memory?
• RQ2 Is it possible to adapt the single-threshold greedy approach—successful in the monotone case—to the non-monotone setting in streaming?
• RQ3 Can existing high-quality offline submodular maximization algorithms be effectively leveraged in the streaming model via post-processing?
• RQ4 What is the optimal approximation ratio achievable in the streaming model for non-monotone submodular maximization, given poly(k, 1/ε) space?
• RQ5 Can a deterministic, combinatorial streaming algorithm with fast update time be designed for non-monotone submodular maximization?

Key findings

• The proposed algorithm achieves an approximation ratio of α/(1+α) − ε, where α is the approximation ratio of the offline algorithm used in post-processing.
• Using an exact (exponential-time) offline algorithm for post-processing yields a (1/2 − ε)-approximation, matching the known inapproximability lower bound of 1/2 for this problem in the streaming model.
• With the state-of-the-art polynomial-time offline algorithm (α = 0.385), the method achieves a 0.2779-approximation, improving over the previous best of 0.1715.
• The algorithm uses only O(k/ε²) memory and has a fast update time of O((log k + log(1/α))/ε²) per element, making it efficient for large-scale data.
• The algorithm is deterministic and combinatorial, which is rare in non-monotone submodular maximization, and avoids reliance on continuous relaxations or complex distributions.
• The analysis shows that E[f(S1,1 ∪ (O1 ∩ U))] ≥ (1 − 3ε)f(OPT) − κb, and that the final solution achieves E[max{f(S1,1), f(T)}] ≥ (α/(1+α) − 3ε)f(OPT), proving the main approximation guarantee.