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Subquadratic Submodular Function Minimization [article]

Deeparnab Chakrabarty and Yin Tat Lee and Aaron Sidford and Sam Chiu-wai Wong
2016 arXiv   pre-print
Submodular function minimization (SFM) is a fundamental discrete optimization problem which generalizes many well known problems, has applications in various fields, and can be solved in polynomial time  ...  The main contribution of this paper are subquadratic SFM algorithms.  ...  Part of this work was done while the first three authors were visiting the Hausdorff Research Institute for Mathematics in Bonn for the Workshop on Submodularity, and the last three authors were visiting  ... 
arXiv:1610.09800v1 fatcat:m52zt4wccvaibn5ku4vkuyvlzm

It's all a matter of degree: Using degree information to optimize multiway joins [article]

Manas Joglekar, Christopher Re
2015 arXiv   pre-print
We say that a join can be processed in subquadratic time if x < 2.  ...  In the serial setting, the data complexity of join processing can be expressed as a function O(^x + ) in terms of input size and output size in which x depends on the query.  ...  In addition, whenever m A happens to be a submodular function over A, m-width is guaranteed to be ≤ submodular width.  ... 
arXiv:1508.01239v7 fatcat:vfqd6dt2d5f5bffvmagx6kfvxu

Graph cuts with many-pixel interactions: Theory and applications to shape modelling

Daniel Freedman, Matthew W. Turek
2010 Image and Vision Computing  
Many problems in computer vision can be posed in terms of energy minimization, where the relevant energy function models the interactions of many pixels.  ...  In this paper, we derive a set of sufficient conditions under which energies which are functions of discrete binary variables may be minimized using graph cut techniques.  ...  Submodularity A well known fact from the theory of combinatorial optimization is that the class of submodular functions can be optimized in polynomial time [18] .  ... 
doi:10.1016/j.imavis.2009.07.006 fatcat:7sxxuhvcwbbkzliccirmvnnyvm

Quantum and Classical Algorithms for Approximate Submodular Function Minimization [article]

Yassine Hamoudi, Patrick Rebentrost, Ansis Rosmanis, Miklos Santha
2020 arXiv   pre-print
Submodular functions are set functions mapping every subset of some ground set of size n into the real numbers and satisfying the diminishing returns property.  ...  Submodular minimization is an important field in discrete optimization theory due to its relevance for various branches of mathematics, computer science and economics.  ...  Introduction Submodular Minimization A submodular function F is a function mapping every subset of some finite set V of size n into the real numbers and satisfying the diminishing returns property: for  ... 
arXiv:1907.05378v2 fatcat:jj2fjffoabbilmyprx2awdhaj4

Computing exact minimum cuts without knowing the graph [article]

Aviad Rubinstein, Tselil Schramm, S. Matthew Weinberg
2019 arXiv   pre-print
Our oracle model is inspired by the submodular function minimization problem: on query S ⊂ V, the oracle returns the size of the cut between S and V ∖ S.  ...  A submodular function is symmetric if f (S) = f (S) for all S. ∅ and [n] are always minimizers of a symmetric submodular function, so the "symmetric submodular function minimization problem" is to find  ...  While graph cuts are indeed a very special case of submodular functions, can any of the ideas from our work be used in randomized algorithms for a broader class of submodular function minimization?  ... 
arXiv:1711.03165v2 fatcat:f2o5zvipifepxguuoeldzwbjma

Near-optimal Approximate Discrete and Continuous Submodular Function Minimization [article]

Brian Axelrod, Yang P. Liu, Aaron Sidford
2019 arXiv   pre-print
In this paper we provide improved running times and oracle complexities for approximately minimizing a submodular function.  ...  Our main result is a randomized algorithm, which given any submodular function defined on n-elements with range [-1, 1], computes an ϵ-additive approximate minimizer in Õ(n/ϵ^2) oracle evaluations with  ...  Theorem 2 (Pseudopolynomial submodular function minimization).  ... 
arXiv:1909.00171v1 fatcat:npzhhspxsvg6jkye67zt5yiytm

Learning with Submodular Functions: A Convex Optimization Perspective [article]

Francis Bach
2013 arXiv   pre-print
In particular, we show how submodular function minimization is equivalent to solving a wide variety of convex optimization problems.  ...  This allows the derivation of new efficient algorithms for approximate and exact submodular function minimization with theoretical guarantees and good practical performance.  ...  The author would like to thank Rodolphe Jenatton, Armand Joulin, Simon Lacoste-Julien, Julien Mairal and Guillaume Obozinski for discussions related to submodular functions and convex optimization.  ... 
arXiv:1111.6453v2 fatcat:qsbgrxoot5f7jhss4otffr3izy

Learning with Submodular Functions: A Convex Optimization Perspective

Francis Bach
2013 Foundations and Trends® in Machine Learning  
The author would like to thank Thibaut Horel, Stefanie Jegelka, Rodolphe Jenatton, Armand Joulin, Simon Lacoste-Julien, Julien Mairal and Guillaume Obozinski for discussions related to submodular functions  ...  -Submodular function minimization: In Chapter 10, we present various approaches to submodular function minimization.  ...  (Lattice of minimizers of submodular functions) Let F be a submodular function such that F (∅) = 0.  ... 
doi:10.1561/2200000039 fatcat:kk7w6zsnsnbp3eoa6b5ol3bxbq

Decomposable Submodular Function Minimization via Maximum Flow [article]

Kyriakos Axiotis, Adam Karczmarz, Anish Mukherjee, Piotr Sankowski, Adrian Vladu
2021 arXiv   pre-print
This paper bridges discrete and continuous optimization approaches for decomposable submodular function minimization, in both the standard and parametric settings.  ...  We achieve this by providing a simple iterative method which can optimize to high precision any convex function defined on the submodular base polytope, provided we can efficiently minimize it on the base  ...  Cohen for discussions on the potential of second order methods for submodular minimization, which motivated parts of this project.  ... 
arXiv:2103.03868v1 fatcat:uojo7yrkwfegtlahrzbqmer2n4

Reinforcement Learning with Dynamic Boltzmann Softmax Updates

Ling Pan, Qingpeng Cai, Qi Meng, Wei Chen, Longbo Huang
2020 Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence  
Value function estimation is an important task in reinforcement learning, i.e., prediction. The Boltzmann softmax operator is a natural value estimator and can provide several benefits.  ...  Experimental results on GridWorld show that the DBS operator enables better estimation of the value function, which rectifies the convergence issue of the softmax operator.  ...  With this choice of Γ, we denote the norm (2) as Ω S 2 (w). prox Ω is more expensive in this case: can be computed using a divide-andconquer strategy involving a sequence of submodular function minimization  ... 
doi:10.24963/ijcai.2020/272 dblp:conf/ijcai/SankaranBB20 fatcat:xhlvsotuprbc7gbz4mw2dguhum

Identifying Groups of Strongly Correlated Variables through Smoothed Ordered Weighted L 1 -norms

Raman Sankaran, Francis R. Bach, Chiranjib Bhattacharyya
2017 International Conference on Artificial Intelligence and Statistics  
In this paper we take a submodular perspective and show that OWL can be posed as the Lovász extension of a suitably defined submodular function.  ...  The submodular perspective not only explains the groupwise constant behavior of OWL, but also suggests alternatives.  ...  They studied the algorithms to compute the norm Ω and its proximal operator using submodular function minimization, which scales typically as O(d 2 log d), using the divide-andconquer algorithm, which  ... 
dblp:conf/aistats/SankaranBB17 fatcat:lgripujedndktlflig4awsplxe

Breaking the Quadratic Barrier for Matroid Intersection [article]

Joakim Blikstad, Jan van den Brand, Sagnik Mukhopadhyay, Danupon Nanongkai
2021 arXiv   pre-print
The goal is to minimize the number of queries. Beating the existing Õ(n^2) bound, known as the quadratic barrier, is an open problem that captures the limits of techniques from two lines of work.  ...  Proving an n 1+Ω(1) lower bound or subquadratic upper bound for, e.g., finding the minimizer of a submodular function or the non-trivial minimizer of a symmetric submodular function.  ...  Related problems are those for minimizing submodular functions.  ... 
arXiv:2102.05548v2 fatcat:rez2h7vtbngrjjrm2bd3sa4rfu

Front Matter, Table of Contents, Preface, Conference Organization

Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, Stefano Leonardi, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming  
Analysis Adam Kurpisz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79:1-79:15 Dynamic Time Warping in Strongly Subquadratic  ...  . . . . 83:1-83:14 Short Proofs Are Hard to Find Ian Mertz, Toniann Pitassi, and Yuanhao Wei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84:1-84:16 A Tight Approximation for Submodular  ... 
doi:10.4230/lipics.icalp.2019.0 dblp:conf/icalp/X19 fatcat:wmdp42kmxnc2lospxghpptbwvi

Locating tree-shaped facilities using the ordered median objective

J. Puerto, A. Tamir
2004 Mathematical programming  
In the tactical model, there is an explicit bound L on the length of the subtree, and the goal is to select a subtree of size L, which minimizes the above transportation cost function.  ...  We also prove submodularity properties for the strategic problem. These properties allow us to solve the discrete strategic version in strongly polynomial time.  ...  Specifically, we will formulate this discrete model as a minimization problem of a submodular function over a lattice.  ... 
doi:10.1007/s10107-004-0547-2 fatcat:qvg5bpicjngn7bwvfbn3lhrpby

Scalable Vaccine Distribution in Large Graphs given Uncertain Data

Yao Zhang, B. Aditya Prakash
2014 Proceedings of the 23rd ACM International Conference on Conference on Information and Knowledge Management - CIKM '14  
Then Q(S) is a submodular because the linear combination of submodular functions is still a submodular function.  ...  Interestingly, Q(S) is a submodular function, while δG i (S) is not submodular [46] . Lemma 2. (Submodularity of Q(S)) Q(S) is a submodu- lar function. Proof.  ... 
doi:10.1145/2661829.2662088 dblp:conf/cikm/ZhangP14 fatcat:2ph62bjymjhvbb3dw3ibvxsbye
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