Approximability of Monotone Submodular Function Maximization under Cardinality and Matroid Constraints in the Streaming Model

by   Chien-Chung Huang, et al.

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 much 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 1-1/e 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+√(2)+ε requires Ω(n/K^2) 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 K/2K-1+ε requires Ω(n/K) space for any ε>0, where K is the rank of the given matroid. In addition, we give streaming algorithms when we only have a weak oracle with which we can only evaluate function values on feasible sets. Specifically, we show weak-oracle streaming algorithms for cardinality and matroid constraints with approximation ratios K/2K-1 and 1/2, 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 K/2K-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.


page 1

page 2

page 3

page 4


Multi-Pass Streaming Algorithms for Monotone Submodular Function Maximization

We consider maximizing a monotone submodular function under a cardinalit...

Improved Multi-Pass Streaming Algorithms for Submodular Maximization with Matroid Constraints

We give improved multi-pass streaming algorithms for the problem of maxi...

FPT-Algorithms for the l-Matchoid Problem with Linear and Submodular Objectives

We design a fixed-parameter deterministic algorithm for computing a maxi...

Submodular Streaming in All its Glory: Tight Approximation, Minimum Memory and Low Adaptive Complexity

Streaming algorithms are generally judged by the quality of their soluti...

The One-way Communication Complexity of Submodular Maximization with Applications to Streaming and Robustness

We consider the classical problem of maximizing a monotone submodular fu...

Quick Streaming Algorithms for Maximization of Monotone Submodular Functions in Linear Time

We consider the problem of monotone, submodular maximization over a grou...

Streaming Algorithms for Cardinality-Constrained Maximization of Non-Monotone Submodular Functions in Linear Time

For the problem of maximizing a nonnegative, (not necessarily monotone) ...

Please sign up or login with your details

Forgot password? Click here to reset