A Note on Monotone Submodular Maximization with Cardinality Constraint

1 Introduction

A set function $f : 2^{E} \to R^{+}$ defined on ground $E$ of size $n$ is submodular, if inequality $f (S) + f (T) \geq f (S \cup T) + f (S \cap T)$ holds for any two subsets $S, T \subseteq E$ . It is monotone non-decreasing if $f (S) \leq f (T)$ holds for any two sets $S \subseteq T \subseteq E$ . In this paper we consider the classic problem of maximizing a monotone submodular function subject to a cardinality constraint $| S | \leq k$ , i.e.,

max {f (S) ∣ ∣ S \subseteq E, | S | \leq k} .

Our result is stated as following.

Theorem 1.

There exists an $(1 - 1 / e - ε)$ -approximate algorithm for the cardinality constrained monotone submodular maximization problem, which makes $O (n / ε)$ queries in total.

This result mainly relies on the algorithm in [1], together with that in [2].

2 Algorithm with linear query complexity

Similar to [2], Algorithm $1$ 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 $⌈ log (4 k) ⌉$ sets $V_{i} (i \in [⌈ log (4 k) ⌉])$ , where $V_{i}$ stores the elements with weight belonging to interval $[{max}_{e} f (e) / 2^{i - 1}, {max}_{e} f (e) / 2^{i})$ , elements stored in $V_{⌈ log (4 k) ⌉}$ have a weight less than $\frac{{max}_{e} f (e)}{4 k}$ . Instead of considering the element in the candidate solution with minimum weight, each time we select an arbitrary element in $V_{ι}$ , where $ι$ represents the largest index $i$ such that $V_{i}$ is non-empty. It can be verified that this implementation has a running time of $O (n log log k)$ . We do not attempt to obtain the most efficient implementation of the algorithm, instead we focus on the query complexity.

1 Initialization:

S, T \leftarrow \emptyset

V_{i} \leftarrow \emptyset (\forall i \in [⌈ log (4 k) ⌉])

w_{0} \leftarrow {max}_{e \in E} f (e)

2 for $e \in E$ do

w (e) \leftarrow f (S \cup T + e) - f (S \cup T)

ℓ \leftarrow ⌊ log (w_{0} / w_{e}) ⌋

ι \leftarrow max {i | V_{i} \neq \emptyset}

e^{†} \leftarrow

an arbitrary element in

V_{ι}

5 if $| S | < k$ then

S \leftarrow S + e

V_{ℓ} \leftarrow V_{ℓ} + e

7 else if $w (e) \geq 2 w (e^{†})$ then

S \leftarrow S + e - e^{†}

T \leftarrow T + e

V_{ℓ} \leftarrow V_{ℓ} + e

V_{ι} \leftarrow V_{ι} - e^{†}

τ \leftarrow \frac{13 f (S)}{k}

12 while $τ \geq \frac{f (S)}{e k}$ do

13 for $e \in E$ do

14 if $f (U + e) - f (U) \geq τ$ and $| U | \leq k$ then

U \leftarrow U + e

τ \leftarrow τ - \frac{ε f (S)}{k}

return $U$

Algorithm 1 A Linear Query Complexity Algorithm for Cardinality Constraint

Lemma 2 ([1]).

For any set $X$ , we use $w (X)$ to denote the total weights of elements in $X$ . Then

w (O P T) \leq \frac{9}{2} w (S) + \frac{f (O P T)}{2} .

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 $S_{e}$ be the value of $S$ before considering element $e$ . For each $e \in O P T$ , the element is charged $w (e)$ if it is added into $S_{e}$ . Otherwise $w (e) < 4 w (e^{'}) + \frac{w_{0}}{2 k}$ holds for $\forall e^{'} \in S_{e}$ , we charge $w (e)$ to an uncharged element $e^{'} \in S_{e}$ . We remark that there always exists an uncharged element in $S_{e}$ , since there are $k$ elements in $O P T$ and hence there are at most $k - 1$ charged elements in $S_{e}$ . We redistribute the charge as follows. When element $e$ is added into $S_{e}$ while $e^{†}$ is removed, the charge on $e^{†}$ is transferred to $e$ .

Note that for any $e \in E$ , initially it is charged at most $4 w (e) + \frac{w_{0}}{2 k}$ , the amount of charge obtained from transfer is no more than $w (e) / 2$ . In addition, the element will not be charged again after it is added into the candidate set. Hence we have

\sum e \in O P T w (e) \leq \sum e \in S [(4 w (e) + \frac{w_{0}}{2 k}) + \frac{w (e)}{2}] = \frac{9}{2} w (S) + \frac{w_{0}}{2} \leq \frac{9}{2} w (S) + \frac{f (O P T)}{2} .

The proof is complete. ∎

Lemma 3 ([1]).

For set $S$ obtained in the first phase,

f (S) \geq \frac{f (O P T)}{13} .

Proof.

According to Lemma $11$ in [1], we know that

f (O P T) \leq 2 f (S) + w (O P T) .

(1)

In addition, we have the following inequality from Lemma $10$ in [1],

w (S) \leq f (S) .

(2)

Combining (1)-(2) with Lemma 2, the proof is complete. ∎

Theorem 4 ([2]).

For set $U$ returned by Algorithm 1, we have

f (U) \geq (1 - 1 / e - ε) \cdot f (O P T) .

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.

A Note on Monotone Submodular Maximization with Cardinality Constraint

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Altitude Terrain Guarding and Guarding Uni-Monotone Polygons

Efficient Algorithms for Monotone Non-Submodular Maximization with Partition Matroid Constraint

Sensitivity Analysis of Submodular Function Maximization

Non-monotone Submodular Maximization in Exponentially Fewer Iterations

1 Introduction

Theorem 1.

2 Algorithm with linear query complexity

Lemma 2 ([1]).

Proof.

Lemma 3 ([1]).

Proof.

Theorem 4 ([2]).

Proof.

Acknowledgement

References

A Note on Monotone Submodular Maximization with Cardinality Constraint

Related Research

On the Complexity of Dynamic Submodular Maximization

Towards Practical Constrained Monotone Submodular Maximization

A polynomial lower bound on adaptive complexity of submodular maximization

Altitude Terrain Guarding and Guarding Uni-Monotone Polygons

Efficient Algorithms for Monotone Non-Submodular Maximization with Partition Matroid Constraint

Sensitivity Analysis of Submodular Function Maximization

Non-monotone Submodular Maximization in Exponentially Fewer Iterations

1 Introduction

Theorem 1.

2 Algorithm with linear query complexity

Lemma 2 ([1]).

Proof.

Lemma 3 ([1]).

Proof.

Theorem 4 ([2]).

Proof.

Acknowledgement

References