## 1 Introduction

A set function defined on ground of size is *submodular*, if inequality holds for any two subsets . It is *monotone non-decreasing* if holds for any two sets . In this paper we consider the classic problem of maximizing a monotone submodular function subject to a cardinality constraint , i.e.,

Our result is stated as following.

###### Theorem 1.

There exists an -approximate algorithm for the cardinality constrained monotone submodular maximization problem, which makes queries in total.

## 2 Algorithm with linear query complexity

Similar to [2], Algorithm consists of two phases—a *preprocessing procedure* and a *refined threshold decreasing procedure*. We use the algorithm in [1] as the first phase, while the second phase is the same as that in [2]. To reduce the running time of the implementation, we maintain sets , where stores the elements with weight belonging to interval , elements stored in have a weight less than
. Instead of considering the element in the candidate solution with minimum weight, each time we select an arbitrary element in , where represents the largest index such that is non-empty. It can be verified that this implementation has a running time of . We do not attempt to obtain the most efficient implementation of the algorithm, instead we focus on the query complexity.

###### Lemma 2 ([1]).

For any set , we use to denote the total weights of elements in . Then

###### Proof.

The proof of this lemma follows from the charging scheme in [1]. We simplify the arguments of [1] in the context of cardinality constraint, and slightly modify the analysis to show the correctness of preprocessing phase. Let be the value of before considering element . For each , the element is charged if it is added into . Otherwise holds for , we charge to an uncharged element . We remark that there always exists an uncharged element in , since there are elements in and hence there are at most charged elements in . We redistribute the charge as follows. When element is added into while is removed, the charge on is transferred to .

Note that for any , initially it is charged at most , the amount of charge obtained from transfer is no more than . In addition, the element will not be charged again after it is added into the candidate set. Hence we have

The proof is complete. ∎

###### Lemma 3 ([1]).

For set obtained in the first phase,

###### Proof.

###### Theorem 4 ([2]).

For set returned by Algorithm 1, we have

###### Proof.

The proof is similar to the proof of Theorem 1 in [2]. ∎

## Acknowledgement

The author would like to thank the anonymous reviewer for making this note possible.

## References

- [1] Amit Chakrabarti and Sagar Kale. Submodular maximization meets streaming: Matchings, matroids, and more. In IPCO, pages 210–221, 2014.
- [2] Wenxin Li and Ness Shroff. Efficient algorithms and lower bound for submodular maximization. arXiv preprint arXiv:1804.08178, 2018.