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1. Maximizing Submodular Functions under Submodular Constraints

   Padmanabhan, Madhavan R; Zhu, Yanhui; Basu, Samik; Pavan A. (July 2023, Uncertainty in Artificial Intelligence, {UAI} 2023)

   Evans, Robin; Shpitser, Ilya (Ed.)

   We consider the problem of maximizing submodular functions under submodular constraints by formulating the problem in two ways: \SCSKC and \DiffC. Given two submodular functions $$f$$ and $$g$$ where $$f$$ is monotone, the objective of \SCSKC problem is to find a set $$S$$ of size at most $$k$$ that maximizes $f(S)$ under the constraint that $$g(S)\leq \theta$$, for a given value of $$\theta$$. The problem of \DiffC focuses on finding a set $$S$$ of size at most $$k$$ such that $h(S) = f(S)-g(S)$$ is maximized. It is known that these problems are highly inapproximable and do not admit any constant factor multiplicative approximation algorithms unless NP is easy. Known approximation algorithms involve data-dependent approximation factors that are not efficiently computable. We initiate a study of the design of approximation algorithms where the approximation factors are efficiently computable. For the problem of \SCSKC, we prove that the greedy algorithm produces a solution whose value is at least $$(1-1/e)f(\OPT) - A$, where $$A$$ is the data-dependent additive error. For the \DiffC problem, we design an algorithm that uses the \SCSKC greedy algorithm as a subroutine. This algorithm produces a solution whose value is at least $$(1-1/e)h(\OPT)-B$, where $$B$$ is also a data-dependent additive error. A salient feature of our approach is that the additive error terms can be computed efficiently, thus enabling us to ascertain the quality of the solutions produced. 

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2. Distributed Maximization of Submodular and Approximately Submodular Functions

   https://doi.org/10.1109/CDC42340.2020.9304492  

   Ye, Lintao; Sundaram, Shreyas (December 2020, 2020 59th IEEE Conference on Decision and Control (CDC))
3. Submodular + Concave

   Mitra, Siddharth; Feldman, Moran; Karbasi, Amin (January 2021, Neural Information Processing Systems)
4. Optimal Algorithms for Continuous Non-monotone Submodular and DR-Submodular Maximization

   Niazadeh, Rad; Roughgarden, Tim; Wang, Joshua R. (December 2018, 31st Annual Conference on Neural Information Processing Systems (NeurIPS), December 2018.)

   This paper considers the problems of maximizing a continuous non-monotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. The main result is the first 1/2-approximation algorithm for continuous submodular function maximization; this approximation factor of 1/2 is the best possible for algorithms that only query the objective function at polynomially many points. For the special case of DR-submodular maximization, i.e. when the submodular functions are also coordinate-wise concave along all coordinates, we provide a different 1 2-approximation algorithm that runs in quasi-linear time. 

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5. Optimal Algorithms for Continuous Non-monotone Submodular and DR-Submodular Maximization

   Niazadeh, Rad; Roughgarden, Tim; Wang, Joshua R. (April 2020, Journal of machine learning research)

   Kraus, Andreas (Ed.)

   In this paper we study the fundamental problems of maximizing a continuous nonmonotone submodular function over the hypercube, both with and without coordinate-wise concavity. This family of optimization problems has several applications in machine learning, economics, and communication systems. Our main result is the first 1 2 -approximation algorithm for continuous submodular function maximization; this approximation factor of 1 2 is the best possible for algorithms that only query the objective function at polynomially many points. For the special case of DR-submodular maximization, i.e. when the submodular function is also coordinate-wise concave along all coordinates, we provide a different 1 2 -approximation algorithm that runs in quasi-linear time. Both these results improve upon prior work (Bian et al., 2017a,b; Soma and Yoshida, 2017). Our first algorithm uses novel ideas such as reducing the guaranteed approximation problem to analyzing a zero-sum game for each coordinate, and incorporates the geometry of this zero-sum game to fix the value at this coordinate. Our second algorithm exploits coordinate-wise concavity to identify a monotone equilibrium condition sufficient for getting the required approximation guarantee, and hunts for the equilibrium point using binary search. We further run experiments to verify the performance of our proposed algorithms in related machine learning applications. 

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