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 $$(0.5-\varepsilon )$$ -approximate solution in $$O(K\varepsilon ^{-1})$$ 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 $$(0.4-\varepsilon )$$ -approximation algorithm for the knapsack-constrained problem, using space that is a polynomial of K and $$\varepsilon $$ . This improves on the previous best ratio of $$0.363-\varepsilon $$ 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.
Highlights
A set function f : 2E → R+ on a ground set E is called submodular if it satisfies the diminishing marginal return property, i.e., for any subsets S ⊆ T E and e ∈ E \ T, we have f (S ∪ {e}) − f (S) ≥ f (T ∪ {e}) − f (T )
Many of the aforementioned applications can be formulated as the maximization of a monotone submodular function under a knapsack constraint
We are given a monotone submodular function f : 2E → R+, a size function c : E → N, and an integer K ∈ N, where N denotes the set of positive integers
Summary
A set function f : 2E → R+ on a ground set E is called submodular if it satisfies the diminishing marginal return property, i.e., for any subsets S ⊆ T E and e ∈ E \ T , we have f (S ∪ {e}) − f (S) ≥ f (T ∪ {e}) − f (T ). The difficulty in improving this algorithm lies in the following case: A new arriving item that is relatively large in size, passes the marginal-ratio threshold, and is part of OPT, but its addition would cause the current set to exceed the capacity K. In this case, we are forced to throw it away, but in doing so, we are unable to bound the ratio of the function value of the current set against that of OPT properly. We remark that the proofs of some lemmas and theorems are omitted due to the page limitation, which can be found in the full version of this paper
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