Improved Streaming Algorithms for Maximizing Monotone Submodular Functions Under a Knapsack Constraint

Abstract

In this paper, we consider the problem of maximizing a monotone submodular function subject to a knapsack constraint in a streaming setting. In such a setting, elements arrive sequentially and at any point in time, and the algorithm can store only a small fraction of the elements that have arrived so far. For the special case that all elements have unit sizes (i.e., the cardinality-constraint case), one can find a (0.5 − ε)-approximate solution in O(Kε −1 ) space, where K is the knapsack capacity (Badanidiyuru et al. KDD 2014). The approximation ratio is recently shown to be optimal (Feldman et al. STOC 2020). In this work, we propose a (0.4 − ε)-approximation algorithm for the knapsack-constrained problem, using space that is a polynomial of K and ε. This improves on the previous best ratio of 0.363 − ε with space of the same order. Our algorithm is based on a careful combination of various ideas to transform multiple-pass streaming algorithms into a single-pass one.

Streaming Algorithms for Non-Submodular Maximization on the Integer Lattice

“…The extensive applications of submodular function in practical problems have also promoted the development of submodular theory [10,11] . Numerous studies on submodular functions have paid close attention to the case of set functions [12][13][14][15][16] . It is known that we can use a simple greedy algorithm to approximately maximize a submodular set function in polynomial time with approximation guarantees.…”

One‐pass streaming algorithm for monotone lattice submodular maximization subject to a cardinality constraint

“…However, many works have been done for the maximization of set submodular functions. Among them streaming algorithms 8‐12 have been well developed since it appears in 2014 8 …”

“…However, many works have been done for the maximization of set submodular functions. Among them streaming algorithms [8][9][10][11][12] have been well developed since it appears in 2014. 8 In the present article, we focus on the streaming model of maximization of monotone nonnegative lattice submodular function with a cardinality on the integer lattice.…”