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Submodular Function Maximization over Distributive and Integer Lattices [article]

Corinna Gottschalk, Britta Peis
2016 arXiv   pre-print
In this paper, we present a deterministic 1/3-approximation for maximizing a submodular function on a bounded integer lattice {0, ..., C}^n using a Double Greedy framework.  ...  Recently, several advances have been made for the more general case of submodular functions on the integer lattice.  ...  refer to (2) as Submodular maximization on a bounded integer lattice (SMBIL).  ... 
arXiv:1505.05423v2 fatcat:wfotmsgaqrckjnhowi7fda3zua

Randomized Algorithms for Monotone Submodular Function Maximization on the Integer Lattice [article]

Alberto Schiabel and Vyacheslav Kungurtsev and Jakub Marecek
2021 arXiv   pre-print
In this work, we consider the problem of maximizing a monotone submodular function on the bounded integer lattice subject to a cardinality constraint.  ...  In particular, we focus on maximizing DR-submodular functions, i.e., functions defined on the integer lattice that exhibit the diminishing returns property.  ...  In this work, we focus on monotone submodular function maximization on the integer lattice subject to a cardinality constraint. More precisely, consider the maximization problem: max f (x) s.t.  ... 
arXiv:2111.10175v1 fatcat:ujpiqmlq4rglhj527wvbub5z54

Optimal Budget Allocation: Theoretical Guarantee and Efficient Algorithm

Tasuku Soma, Naonori Kakimura, Kazuhiro Inaba, Ken-ichi Kawarabayashi
2014 International Conference on Machine Learning  
We first show that this problem and its much more general form fall into a general setting; namely the monotone submodular function maximization over integer lattice subject to a knapsack constraint.  ...  This problem can be viewed as the well-known influence maximization problem with budget constraints.  ...  Let f be a monotone submodular function over integer lattice.  ... 
dblp:conf/icml/SomaKIK14 fatcat:avd7wfxebrg47apyalpv7ysrhi

Subspace Selection via DR-Submodular Maximization on Lattices

So Nakashima, Takanori Maehara
2019 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
Then, we introduce a new class of functions on lattices, directional DR-submodular functions, to characterize the approximability of problems.  ...  set of subspaces forms a lattice, then formulate the problems as optimization problems on lattices.  ...  Recall definition (1.4) of the DR-submodularity on the integer lattice.  ... 
doi:10.1609/aaai.v33i01.33014618 fatcat:2we5whxzenb5nfc5my2ibx6uuu

Non-monotone DR-Submodular Function Maximization [article]

Tasuku Soma, Yuichi Yoshida
2016 arXiv   pre-print
We consider non-monotone DR-submodular function maximization, where DR-submodularity (diminishing return submodularity) is an extension of submodularity for functions over the integer lattice based on  ...  Maximizing non-monotone DR-submodular functions has many applications in machine learning that cannot be captured by submodular set functions.  ...  Related work As mentioned above, there have been many efforts to maximize submodular functions on the integer lattice.  ... 
arXiv:1612.00960v1 fatcat:5pxxlfywkjffji6kpg7hxgjrte

Fast Maximization of Non-Submodular, Monotonic Functions on the Integer Lattice [article]

Alan Kuhnle, J. David Smith, Victoria G. Crawford, My T. Thai
2018 arXiv   pre-print
The optimization of submodular functions on the integer lattice has received much attention recently, but the objective functions of many applications are non-submodular.  ...  We provide two approximation algorithms for maximizing a non-submodular function on the integer lattice subject to a cardinality constraint; these are the first algorithms for this purpose that have polynomial  ...  DR ratio γ d = 1, see Section 3) functions on the integer lattice.  ... 
arXiv:1805.06990v1 fatcat:2ckcwt5funaxvohgjgt7vhn5kq

Non-Monotone DR-Submodular Function Maximization

Tasuku Soma, Yuichi Yoshida
2017 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
We consider non-monotone DR-submodular function maximization, where DR-submodularity (diminishing return submodularity) is an extension of submodularity for functions over the integer lattice based on  ...  Maximizing non-monotone DR-submodular functions has many applications in machine learning that cannot be captured by submodular set functions.  ...  Related work As mentioned above, there have been many efforts to maximize submodular functions on the integer lattice.  ... 
doi:10.1609/aaai.v31i1.10653 fatcat:njqkdlgp7bdrvbzk7bjwj6v5c4

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

An Bian, Kfir Yehuda Levy, Andreas Krause, Joachim M. Buhmann
2017 Neural Information Processing Systems  
In this work we study the problem of maximizing non-monotone continuous DRsubmodular functions under general down-closed convex constraints.  ...  DR-submodularity captures a subclass of non-convex functions that enables both exact minimization and approximate maximization in polynomial time.  ...  This research was partially supported by ERC StG 307036, by the Max Planck ETH Center for Learning Systems, and by the ETH Zürich Postdoctoral Fellowship program.  ... 
dblp:conf/nips/BianL0B17 fatcat:wcelfk4fbvcrlhdogxvbkwal6q

On the Caratheodory rank of polymatroid bases [article]

Dion Gijswijt, Guus Regts
2010 arXiv   pre-print
In this paper we prove that the Carath\'eodory rank of the set of bases of a (poly)matroid is upper bounded by the cardinality of the ground set.  ...  A submodular function f : P(E) → Z is the rank function of a matroid M on E if and only if f is nonnegative, nondecreasing and f (U ) ≤ |U | for every set U ⊆ E.  ...  In his paper on testing membership in matroid polyhedra, Cunningham [4] first asked for an upper bound on the number of different bases needed in a representation of a vector as a nonnegative integer  ... 
arXiv:1003.1079v1 fatcat:p3riouiq7fcl5kz6ljddykyagm

Offline and Online Models of Budget Allocation for Maximizing Influence Spread [article]

Noa Avigdor-Elgrabli, Gideon Blocq, Iftah Gamzu, Ariel Orda
2018 arXiv   pre-print
We establish that any function in this family implies an instance of a monotone submodular function maximization over the integer lattice subject to a knapsack constraint.  ...  This setting extends the secretary problem, and its variant, the submodular knapsack secretary problem.  ...  The social function F (b) = M i=1 f i (b) is a monotone submodular function on the integer lattice. Proof.  ... 
arXiv:1508.01059v3 fatcat:h4eb76sofbcfzj3t6gm6wip7aa

A Generalization of Submodular Cover via the Diminishing Return Property on the Integer Lattice

Tasuku Soma, Yuichi Yoshida
2015 Neural Information Processing Systems  
We consider a generalization of the submodular cover problem based on the concept of diminishing return property on the integer lattice.  ...  We are motivated by real scenarios in machine learning that cannot be captured by (traditional) submodular set functions.  ...  The second author is supported by JSPS Grant-in-Aid for Young Scientists (B) (No. 26730009), MEXT Grant-in-Aid for Scientific Research on Innovative Areas (24106003), and JST, ERATO, Kawarabayashi Large  ... 
dblp:conf/nips/SomaY15 fatcat:5jssj5pognb43fw2qxoyfbhiyy

Maximizing Monotone Submodular Functions over the Integer Lattice [article]

Tasuku Soma, Yuichi Yoshida
2016 arXiv   pre-print
In this paper, we address the problem for functions defined over the integer lattice. Suppose that a non-negative monotone submodular function f:Z_+^n →R_+ is given via an evaluation oracle.  ...  The problem of maximizing non-negative monotone submodular functions under a certain constraint has been intensively studied in the last decade.  ...  Gottschalk and Peis [13] provided a 1/3-approximation algorithm for maximizing a lattice submodular function over a (bounded) integer lattice.  ... 
arXiv:1503.01218v2 fatcat:pxjubzhyfjfrnegchq4c2mxe7m

Continuous DR-submodular Maximization: Structure and Algorithms [article]

An Bian, Kfir Y. Levy, Andreas Krause, Joachim M. Buhmann
2019 arXiv   pre-print
In this work we study the problem of maximizing non-monotone DR-submodular continuous functions under general down-closed convex constraints.  ...  DR-submodularity captures a subclass of non-convex functions that enables both exact minimization and approximate maximization in polynomial time.  ...  This research was partially supported by ERC StG 307036, by the Max Planck ETH Center for Learning Systems, and by the ETH Zürich Postdoctoral Fellowship program.  ... 
arXiv:1711.02515v4 fatcat:by4uo3vitfdj7kdu5n4v6tcoee

Robust Budget Allocation via Continuous Submodular Functions [article]

Matthew Staib, Stefanie Jegelka
2017 arXiv   pre-print
We show that this nonconvex problem can be solved exactly by leveraging connections to continuous submodular functions, and by solving a constrained submodular minimization problem.  ...  The optimal allocation of resources for maximizing influence, spread of information or coverage, has gained attention in the past years, in particular in machine learning and data mining.  ...  Acknowledgements We thank the anonymous reviewers for their helpful suggestions. We also thank MIT Supercloud and the Lincoln Laboratory Supercomputing Center for providing computational resources.  ... 
arXiv:1702.08791v2 fatcat:j4pnb73gu5cdbcvi4wxpzy6ozm

Guaranteed Non-convex Optimization: Submodular Maximization over Continuous Domains [article]

Andrew An Bian, Baharan Mirzasoleiman, Joachim M. Buhmann, Andreas Krause
2019 arXiv   pre-print
Specifically, i) We introduce the weak DR property that gives a unified characterization of submodularity for all set, integer-lattice and continuous functions; ii) for maximizing monotone DR-submodular  ...  Submodular continuous functions are a category of (generally) non-convex/non-concave functions with a wide spectrum of applications.  ...  Acknowledgments The authors would like to thank Martin Jaggi for valuable discussions.  ... 
arXiv:1606.05615v5 fatcat:meh4cdyikfgxzgbuvpdh2eckri
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