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A Nearly-linear Time Algorithm for Submodular Maximization with a Knapsack Constraint [article]

Alina Ene, Huy L. Nguyen
2018 arXiv   pre-print
We consider the problem of maximizing a monotone submodular function subject to a knapsack constraint.  ...  This is the main approach for obtaining nearly-optimal approximation guarantees for important classes of constraints but it leads to Ω(n^2) running times, since evaluating the multilinear extension is  ...  Obtaining near-optimal approximations in nearly-linear time for submodular maximization has been out of reach for all but a cardinality constraint.  ... 
arXiv:1709.09767v3 fatcat:32sjeapg7vaxhpn46zcx6scbdm

A Nearly-Linear Time Algorithm for Submodular Maximization with a Knapsack Constraint

Alina Ene, Huy L. Nguyen, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming  
We consider the problem of maximizing a monotone submodular function subject to a knapsack constraint.  ...  This is the main approach for obtaining nearly-optimal approximation guarantees for important classes of constraints but it leads to Ω(n 2 ) running times, since evaluating the multilinear extension is  ...  Obtaining near-optimal approximations in nearly-linear time for submodular maximization has been out of reach for all but a cardinality constraint.  ... 
doi:10.4230/lipics.icalp.2019.53 dblp:conf/icalp/EneN19 fatcat:g4budbgib5fqra6riemczqofsm

Approximations for Monotone and Non-monotone Submodular Maximization with Knapsack Constraints [article]

Ariel Kulik, Hadas Shachnai, Tami Tamir
2011 arXiv   pre-print
In this paper we consider the problem of maximizing any submodular function subject to d knapsack constraints, where d is a fixed constant.  ...  Formally, we show that, for any non-negative submodular function, an α-approximation algorithm for the continuous relaxation implies a randomized (α - )-approximation algorithm for the discrete problem  ...  [19] studied the problem of maximizing a general submodular function under linear and matroid constraints.  ... 
arXiv:1101.2940v1 fatcat:5ie2mkgsxng3xid6hv4g6l7vqa

Nearly Linear Time Deterministic Algorithms for Submodular Maximization Under Knapsack Constraint and Beyond [article]

Wenxin Li
2020 arXiv   pre-print
Lastly we present nearly linear time algorithms for the intersection of p-system and d knapsack constraint, we achieve approximation ratio of (1/(p+7/4d+1)-ε) for monotone objective and (p/(p+1)(2p+7/4d  ...  In this work, we study the classic submodular maximization problem under knapsack constraints and beyond.  ...  There is an (1/(p + 7 4 d+ 1)− ε)-approximate algorithm for maximizing a non-negative monotone submodular function subject to a p-system and d knapsack constraints, which performs nearly linear number  ... 
arXiv:1804.08178v6 fatcat:47tudonhwzghpahwwx3h4lzraq

Non-monotone submodular maximization under matroid and knapsack constraints

Jon Lee, Vahab S. Mirrokni, Viswanath Nagarajan, Maxim Sviridenko
2009 Proceedings of the 41st annual ACM symposium on Symposium on theory of computing - STOC '09  
In this paper, we give the first constant-factor approximation algorithm for maximizing any non-negative submodular function subject to multiple matroid or knapsack constraints.  ...  Submodular function maximization is a central problem in combinatorial optimization, generalizing many important problems including Max Cut in directed/undirected graphs and in hypergraphs, certain constraint  ...  Our original proof [35] was more complicated -we thank Jan for letting us present this simplified proof.  ... 
doi:10.1145/1536414.1536459 dblp:conf/stoc/LeeMNS09 fatcat:262larfgkfcdnmyogqcpayxd5a

Randomized Strategies for Robust Combinatorial Optimization

Yasushi Kawase, Hanna Sumita
2019 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
As applications, we provide approximation algorithms for a knapsack constraint or a matroid intersection by developing appropriate relaxations and retrievals.  ...  Using our result, we provide approximation algorithms when the objective functions are submodular or correspond to the cardinality robustness for the knapsack problem.  ...  Because there exist (1 − 1/e)-approximation algorithms for maximizing a monotone submodular function under a knapsack constraint (Sviridenko 2004 ) and under a matroid constraint (Calinescu et al. 2007  ... 
doi:10.1609/aaai.v33i01.33017876 fatcat:bu7n5ghbpzbubplrsh3hcgzbjq

Fast Constrained Submodular Maximization: Personalized Data Summarization

Baharan Mirzasoleiman, Ashwinkumar Badanidiyuru, Amin Karbasi
2016 International Conference on Machine Learning  
In addition, knapsacks encode users' constraints including budget or time.  ...  We develop the first practical and FAst coNsTrained submOdular Maximization algorithm, FANTOM, with strong theoretical guarantees.  ...  This research was supported by a Google Faculty Research Award.  ... 
dblp:conf/icml/MirzasoleimanBK16 fatcat:vlefgd7cqjaydpbwo3p4ckxrlq

Non-monotone submodular maximization under matroid and knapsack constraints [article]

Jon Lee, Vahab Mirrokni, Viswanath Nagarjan, Maxim Sviridenko
2009 arXiv   pre-print
In this paper, we give the first constant-factor approximation algorithm for maximizing any non-negative submodular function subject to multiple matroid or knapsack constraints.  ...  In particular, for any constant k, we present a (1 k+2+1 k+ϵ)-approximation for the submodular maximization problem under k matroid constraints, and a (1 5-ϵ)-approximation algorithm for this problem subject  ...  Our original proof [35] was more complicated -we thank Jan for letting us present this simplified proof.  ... 
arXiv:0902.0353v1 fatcat:ayzfv2mygjhc5esgfqwv2vbceu

"Bring Your Own Greedy"+Max: Near-Optimal 1/2-Approximations for Submodular Knapsack [article]

Dmitrii Avdiukhin, Grigory Yaroslavtsev, Samson Zhou
2019 arXiv   pre-print
this problem as a submodular maximization subject to a linear (knapsack) constraint.  ...  Motivated by applications to recommendation systems and other scenarios with query-limited access to vast amounts of data, we propose a new rigorous algorithmic framework for a standard formulation of  ...  There exists an offline algorithm Greedy+Max (Algorithm 1) that gives a 1 /2-approximation for the submodular maximization problem under a knapsack constraint with query complexity and running time O K  ... 
arXiv:1910.05646v1 fatcat:v3df3bdp3bblnbrold5nzx33xy

Approximations for Monotone and Nonmonotone Submodular Maximization with Knapsack Constraints

Ariel Kulik, Hadas Shachnai, Tami Tamir
2013 Mathematics of Operations Research  
In this paper we consider the problem of maximizing any submodular function subject to d knapsack constraints, where d is a fixed constant.  ...  Our approach has a potential of wider applicability, which we demonstrate on the examples of the Generalized Assignment Problem and Maximum Coverage with additional knapsack constraints.  ...  Theorem 2 . 2 15 There is a polynomial time deterministic (1−e −1 −ε)-approximation algorithm for maximum coverage with d knapsack constraints.  ... 
doi:10.1287/moor.2013.0592 fatcat:calt4f635zaehn37pduhxsl4ce

Streaming Submodular Maximization under a k-Set System Constraint [article]

Ran Haba, Ehsan Kazemi, Moran Feldman, Amin Karbasi
2020 arXiv   pre-print
In this paper, we propose a novel framework that converts streaming algorithms for monotone submodular maximization into streaming algorithms for non-monotone submodular maximization.  ...  Together with our proposed reduction, we obtain O(klog k) and O(k^2log k) approximation ratio for submodular maximization subject to the above constraints, respectively.  ...  Both the Fast and FANTOM algorithms are designed to maximize submodular functions under a p-system constraint combined with knapsack constraints.  ... 
arXiv:2002.03352v1 fatcat:zlt5xt6qkffplb2mraqk66mfwe

Submodular Maximization in Clean Linear Time [article]

Wenxin Li, Moran Feldman, Ehsan Kazemi, Amin Karbasi
2022 arXiv   pre-print
We then provide a variant of our deterministic algorithm for the more general knapsack constraint, which is the first linear-time algorithm that achieves 1/2-approximation guarantee for this constraint  ...  In this paper, we provide the first deterministic algorithm that achieves the tight 1-1/e approximation guarantee for submodular maximization under a cardinality (size) constraint while making a number  ...  For every ε > 0, there exists a deterministic ( 1 /2 − ε)-approximation algorithm for Submodular Maximization subject to a Knapsack Constraint (SMK) that uses O(nε −1 log ε −1 ) time.  ... 
arXiv:2006.09327v5 fatcat:v3gv3h5xmvf3pmh466cgq4shv4

Fast Adaptive Non-Monotone Submodular Maximization Subject to a Knapsack Constraint [article]

Georgios Amanatidis, Federico Fusco, Philip Lazos, Stefano Leonardi, Rebecca Reiffenhäuser
2020 arXiv   pre-print
Focusing on these challenges, we revisit the classic problem of maximizing a (possibly non-monotone) submodular function subject to a knapsack constraint.  ...  We present a simple randomized greedy algorithm that achieves a 5.83 approximation and runs in O(n log n) time, i.e., at least a factor n faster than other state-of-the-art algorithms.  ...  The Algorithmic Idea We present and analyze SampleGreedy, a randomized 5.83-approximation algorithm for maximizing a submodular function subject to a knapsack constraint.  ... 
arXiv:2007.05014v1 fatcat:ynt4meyn7jfulen6c6idgc4yc4

Randomized Strategies for Robust Combinatorial Optimization [article]

Yasushi Kawase, Hanna Sumita
2018 arXiv   pre-print
As applications, we provide approximation algorithms for a knapsack constraint or a matroid intersection by developing appropriate relaxations and retrievals.  ...  Using our result, we provide approximation algorithms when the objective functions are submodular or correspond to the cardinality robustness for the knapsack problem.  ...  For the monotone submodular function maximization problem, there exist (1−1/e)-approximation algorithms under a knapsack constraint [44] or a matroid constraint [8, 15] , and there exists a 1/(µ + ǫ  ... 
arXiv:1805.07809v1 fatcat:qxz23p52njfihj4e2glsmmlhiq

Maximizing Nonmonotone Submodular Functions under Matroid or Knapsack Constraints

Jon Lee, Vahab S. Mirrokni, Viswanath Nagarajan, Maxim Sviridenko
2010 SIAM Journal on Discrete Mathematics  
In this paper, we give the first constant-factor approximation algorithm for maximizing any non-negative submodular function subject to multiple matroid or knapsack constraints.  ...  We emphasize that our results are for non-monotone submodular functions. In particular, for any constant k, we present a  ...  Our original proof [35] was more complicated -we thank Jan for letting us present this simplified proof.  ... 
doi:10.1137/090750020 fatcat:fkozckgiyzgtriks4vti5uvbmu
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