# Submodular Function Maximization

@inproceedings{Krause2014SubmodularFM, title={Submodular Function Maximization}, author={Andreas Krause and Daniel Golovin}, booktitle={Tractability}, year={2014} }

Submodularity is a property of set functions with deep theoretical consequences and far– reaching applications. At first glance it appears very similar to concavity, in other ways it resembles convexity. It appears in a wide variety of applications: in Computer Science it has recently been identified and utilized in domains such as viral marketing (Kempe et al., 2003), information gathering (Krause and Guestrin, 2007), image segmentation (Boykov and Jolly, 2001; Kohli et al., 2009; Jegelka and…

## 772 Citations

### Continuous DR-submodular Maximization: Structure and Algorithms

- Computer Science, MathematicsNIPS 2017
- 2017

This work investigates the problem of maximizing non-monotone DR-submodular continuous functions under general down-closed convex constraints by investigating geometric properties that underlie such objectives, and devise two optimization algorithms with provable guarantees that are validated on synthetic and real-world problem instances.

### Guaranteed Non-convex Optimization: Submodular Maximization over Continuous Domains

- Computer Science, MathematicsAISTATS
- 2017

The weak DR property is introduced that gives a unified characterization of submodularity for all set, integer-lattice and continuous functions and for maximizing monotone DR-submodular continuous functions under general down-closed convex constraints, a Frank-Wolfe variant with approximation guarantee, and sub-linear convergence rate are proposed.

### Regularized Submodular Maximization at Scale

- Computer ScienceICML
- 2021

This paper develops the first one-pass streaming algorithm for maximizing a regularized submodular function subject to a $k$-cardinality constraint and develops a first distributed algorithm that returns a solution $S$ with the guarantee that f(S) \geq(\phi^{-2}-\epsilon) \cdot g(OPT)-\ell (OPT)$, where $\phi$ is the golden ratio.

### Robust Submodular Maximization: A Non-Uniform Partitioning Approach

- Computer ScienceICML
- 2017

A new Partitioned Robust (PRo) submodular maximization algorithm that achieves the same guarantee for more general $\tau = o(k)$ and numerically demonstrates the performance of PRo in data summarization and influence maximization.

### Fast and Private Submodular and k-Submodular Functions Maximization with Matroid Constraints

- Mathematics, Computer ScienceICML
- 2020

This paper gives the first $\frac{1}{2}$-approximation algorithm that preserves differential privacy for maximizing monotone $k$-submodular functions subject to matroid constraints and is obtained with an almost linear number of function evaluations.

### Provable Non-Convex Optimization and Algorithm Validation via Submodularity

- Computer ScienceArXiv
- 2019

The role of submodularity in provable non-convex optimization and validation of algorithms is investigated, and efficient approaches to calculate the algorithmic information content of MaxCut algorithms are presented.

### Non-monotone Continuous DR-submodular Maximization: Structure and Algorithms

- Computer Science, MathematicsNIPS
- 2017

This work investigates the problem of maximizing non-monotone DR-submodular continuous functions under general down-closed convex constraints by investigating geometric properties that underlie such objectives, and devise two optimization algorithms with provable guarantees that are validated on synthetic and real-world problem instances.

### Submodular Maximization Through Barrier Functions

- Computer ScienceNeurIPS
- 2020

This paper introduces a novel technique for constrained submodular maximization, inspired by barrier functions in continuous optimization, and proposes a potential function that can be approximately minimized for maximizing a monotone sub modular function subject to the combination of a $k$-matchoid and $\ell$-knapsack constraint.

### Methods in Two Stage Submodular Maximization

- Computer Science
- 2016

This thesis presents a new approach that offers a faster way of computing the maximum of the two stage objective function, and offers empirical evidence for its performance, and expects it to significantly outperform the other current approaches in speed.

## References

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- Computer Science, MathematicsSODA '11
- 2011

A new algorithm for submodular maximization which is based on the idea of simulated annealing is proposed and it is proved that this algorithm achieves improved approximation for two problems: a 0.41-approximation for unconstrained submodul maximization, and a0.325-app approximation subject to a matroid independence constraint.

### Efficient Minimization of Decomposable Submodular Functions

- Mathematics, Computer ScienceNIPS
- 2010

This paper develops an algorithm, SLG, that can efficiently minimize decomposable submodular functions with tens of thousands of variables, and applies it to synthetic benchmarks and a joint classification-and-segmentation task, and shows that it outperforms the state-of-the-art general purpose sub modular minimization algorithms by several orders of magnitude.

### Submodularity Cuts and Applications

- Computer ScienceNIPS
- 2009

This work presents a novel algorithm for maximizing a submodular set function under a cardinality constraint based on a cutting-plane method and implemented as an iterative small-scale binary-integer linear programming procedure.

### Revisiting the Greedy Approach to Submodular Set Function Maximization

- Mathematics, Computer Science
- 2007

This work explores the performance of the greedy algorithm, and a related variant, the locally greedy algorithm in solving submodular function maximization problems, and reinterpret and improves various algorithmic results in the recent literature for problems that are specific instances of maximizing monotone sub modular functions in the presence of an approximate incremental oracle.

### Submodular function maximization via the multilinear relaxation and contention resolution schemes

- MathematicsSTOC '11
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A broadly applicable framework for maximizing linear and submodular functions subject to independence constraints is developed and it is shown that contention resolution schemes are an effective way to round a fractional solution, even when f is non-monotone.

### Constrained Non-monotone Submodular Maximization: Offline and Secretary Algorithms

- Mathematics, Computer ScienceWINE
- 2010

These ideas are extended to give a simple greedy-based constant factor algorithms for non-monotone submodular maximization subject to a knapsack constraint, and for (online) secretary setting subject to uniform matroid or a partition matroid constraint.

### An analysis of approximations for maximizing submodular set functions—I

- MathematicsMath. Program.
- 1978

It is shown that a “greedy” heuristic always produces a solution whose value is at least 1 −[(K − 1/K]K times the optimal value, which can be achieved for eachK and has a limiting value of (e − 1)/e, where e is the base of the natural logarithm.

### Optimal approximation for the submodular welfare problem in the value oracle model

- MathematicsSTOC
- 2008

A randomized continuous greedy algorithm is developed which achieves a (1-1/e)-approximation for the Submodular Welfare Problem in the value oracle model and is shown to have a potential of wider applicability on the examples of the Generalized Assignment Problem and the AdWords Assignment Problem.

### Adaptive Submodularity: Theory and Applications in Active Learning and Stochastic Optimization

- Computer ScienceJ. Artif. Intell. Res.
- 2011

It is proved that if a problem satisfies adaptive submodularity, a simple adaptive greedy algorithm is guaranteed to be competitive with the optimal policy, providing performance guarantees for both stochastic maximization and coverage.

### Adaptive Submodular Optimization under Matroid Constraints

- Mathematics, Computer ScienceArXiv
- 2011

It is proved that a natural adaptive greedy algorithm provides a $1/(p+1)$ approximation for the problem of maximizing an adaptive monotone submodular function subject to matroid constraints, and more generally over arbitrary $p$-independence systems.