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Minimizing a monotone concave function with laminar covering constraints

Mariko Sakashita, Kazuhisa Makino, Satoru Fujishige
2008 Discrete Applied Mathematics  
Given a laminar family F, a demand function d : F → R + , and a monotone concave cost function F : R V + → R + , we consider the problem of finding a minimum-cost Here we do not assume that the cost function  ...  Let V be a finite set with |V | = n. A family F ⊆ 2 V is called laminar if for all two sets X, Y ∈ F, X ∩ Y = ∅ implies X ⊆ Y or X ⊇ Y .  ...  Acknowledgments The present work is supported by a Grant-in-Aid from Ministry of Education, Culture, Sports, Science, and Technology of Japan.  ... 
doi:10.1016/j.dam.2007.04.016 fatcat:cxsflulerrdvfamvqvrvhlifcq

Minimizing a Monotone Concave Function with Laminar Covering Constraints [chapter]

Mariko Sakashita, Kazuhisa Makino, Satoru Fujishige
2005 Lecture Notes in Computer Science  
Given a laminar family F, a demand function d : F → R + , and a monotone concave cost function F : R V + → R + , we consider the problem of finding a minimum-cost x ∈ R V + such that x(X) ≥ d(X) for all  ...  Let V be a finite set with |V | = n. A family F ⊆ 2 V is called laminar if for all two sets X, Y ∈ F, X ∩ Y = ∅ implies X ⊆ Y or X ⊇ Y .  ...  Acknowledgments: The present work is supported by a Grant-in-Aid from Ministry of Education, Culture, Sports, Science and Technology of Japan. 21 The authors would like to express their appreciation  ... 
doi:10.1007/11602613_9 fatcat:gsnejuc4brg6da7lpmzffufgg4

Contents

2008 Discrete Applied Mathematics  
Fujishige Minimizing a monotone concave function with laminar covering constraints 2004 L. Khachiyan, E. Boros, K. Elbassioni and V.  ...  A. Ben-Israel and Y. Levin The Newton Bracketing method for the minimization of convex functions subject to affine constraints 1977 N.S. Kukushkin E. Boros and V.  ... 
doi:10.1016/s0166-218x(08)00287-4 fatcat:n7ujjgi24bey7ieuwq6lquodsa

Deep Submodular Functions: Definitions and Learning

Brian W. Dolhansky, Jeff A. Bilmes
2016 Neural Information Processing Systems  
We define DSFs and situate them within the broader context of classes of submodular functions in relationship both to various matroid ranks and sums of concave composed with modular functions (SCMs).  ...  We propose and study a new class of submodular functions called deep submodular functions (DSFs).  ...  For example, submodular functions can be minimized without constraints in polynomial time [12] even though they lie within a 2 n -dimensional cone in R 2 n .  ... 
dblp:conf/nips/DolhanskyB16 fatcat:t3zsaouy2naenc4xuseitdz7xa

The submodular joint replenishment problem

Maurice Cheung, Adam N. Elmachtoub, Retsef Levi, David B. Shmoys
2015 Mathematical programming  
In this paper, the cost of an order, also known as a joint setup cost, is a monotonically increasing, submodular function over the item types.  ...  Specifically, we show that the laminar case can be solved optimally in polynomial time via a dynamic programming approach.  ...  The research of the second author was partially conducted while he was a student at the Operations Research Center at Massachusetts Institute of Technology and was supported by a National Defense Science  ... 
doi:10.1007/s10107-015-0920-3 fatcat:gjtmzwdpl5c6hfnjszac63tgz4

On Multiplicative Weight Updates for Concave and Submodular Function Maximization

Chandra Chekuri, T.S. Jayram, Jan Vondrak
2015 Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science - ITCS '15  
The framework allows for a simple and modular analysis for a variety of problems involving convex constraints and concave or submodular objective functions.  ...  set function subject to linear packing constraints. 1 We say S is down-closed if A ⊂ B, B ∈ S ⇒ A ∈ S 2 We say that P ⊆ [0, 1] n is solvable if there is an efficient algorithm to optimize any linear function  ...  In particular, a covering constraint of the form h(x) ≥ 1 where h is a concave function can be modeled as g(x) ≤ 1 where g(x) = −h(x) + 2 ≤ 1 is a convex function.  ... 
doi:10.1145/2688073.2688086 dblp:conf/innovations/ChekuriJV15 fatcat:qh5rdgdmtnbu7c6ataszrho3eq

Deep Submodular Functions [article]

Jeffrey Bilmes, Wenruo Bai
2017 arXiv   pre-print
We start with an overview of a class of submodular functions called SCMMs (sums of concave composed with non-negative modular functions plus a final arbitrary modular).  ...  To further motivate our analysis, we provide various special case results from matroid theory, comparing DSFs with forms of matroid rank, in particular the laminar matroid.  ...  Let φ : R → R be a monotone non-decreasing concave functions and χ : R n → R be a monotone non-decreasing concave function with an antitone superdifferential, and define ψ(x) = φ(χ(x)).  ... 
arXiv:1701.08939v1 fatcat:zvbpgq7d2feqrantvl7hi34jum

Polyhedral aspects of Submodularity, Convexity and Concavity [article]

Rishabh Iyer, Jeff Bilmes
2015 arXiv   pre-print
Concave functions composed with modular functions are submodular, and they also satisfy diminishing returns property.  ...  This manuscript provides a more complete picture on the relationship between submodularity with convexity and concavity, by extending many of the results connecting submodularity with convexity to the  ...  For example, a simple cardinality lower bound constraint makes the problem of submodular minimization (even with monotone submodular functions) NP hard without even constant factor approximation guarantees  ... 
arXiv:1506.07329v2 fatcat:253bjrlb7zf55kqkxmkqjtufk4

Provable Variational Inference for Constrained Log-Submodular Models

Josip Djolonga, Stefanie Jegelka, Andreas Krause
2018 Neural Information Processing Systems  
In addition to providing completely tractable and well-understood variational approximations, our approach results in the minimization of a convex upper bound on the log-partition function.  ...  Because the data defining these functions, as well as the decisions made with the computed solutions, are subject to statistical noise and randomness, it is arguably necessary to go beyond computing a  ...  A classical family of submodular functions are set cover functions.  ... 
dblp:conf/nips/DjolongaJ018 fatcat:ag4xpvgvzrh4jnx5o4xckcoyc4

Minimum Cost Source Location Problems with Flow Requirements [chapter]

Mariko Sakashita, Kazuhisa Makino, Satoru Fujishige
2006 Lecture Notes in Computer Science  
κ), where the source location problems are to find a minimum-cost set S ⊆ V in a given graph G = (V, A) with a capacity function u : A → R + such that for each vertex v ∈ V , the connectivity from S to  ...  Moreover, we show that the source location problems with three connectivity requirements are inapproximable within a ratio of c ln D for some constant c, unless every problem in NP has an O(N log log N  ...  Given a finite set V , monotone concave cost functions c v : R + → R + (v ∈ V ), a monotone submodular function f : R V + → R + and a real M , the problem asks for a minimum-cost vector x ∈ R V + that  ... 
doi:10.1007/11682462_70 fatcat:ik3t7xmsjveadbnxdge5lsgqfy

Submodular Functions: Optimization and Approximation

Satoru Iwata
2011 Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)  
For submodular function minimization, the ellipsoid method had long been the only polynomial algorithm until combinatorial strongly polynomial algorithms appeared a decade ago.  ...  problems with submodular costs.  ...  This is a special case of maximizing a monotone submodular function under a cardinality constraint. Matroid Rank Functions.  ... 
doi:10.1142/9789814324359_0173 fatcat:uspd2ma2uzco3kvzqiytgxz45a

Multiwinner Elections with Diversity Constraints [article]

Robert Bredereck, Piotr Faliszewski, Ayumi Igarashi, Martin Lackner, Piotr Skowron
2017 arXiv   pre-print
We develop a model of multiwinner elections that combines performance-based measures of the quality of the committee (such as, e.g., Borda scores of the committee members) with diversity constraints.  ...  We focus on several natural classes of voting rules and diversity constraints.  ...  Clearly, since f is separable, the functions g Y are piecewise-linear and concave. We will construct a mixed ILP with the following non-linear objective function: minimize Y ∈2 L g Y (z Y ).  ... 
arXiv:1711.06527v2 fatcat:bjl2nvzvjbbshpv6mhatr4llve

Discrete convex analysis: A tool for economics and game theory

Kazuo Murota
2016 Journal of Mechanism and Institution Design  
Recently, it has been recognized as a powerful tool for analyzing economic or game models with indivisibilities.  ...  The main feature of discrete convex analysis is the distinction of two convexity concepts, M-convexity and L-convexity, for functions in integer or binary variables, together with their conjugacy relationship  ...  Every unique-selecting M -concave function f : 2 N → R ∪ {−∞} with / 0 ∈ dom f induces a choice function with cardinal monotonicity. Proof.  ... 
doi:10.22574/jmid.2016.12.005 fatcat:e73iapgazbcxljnr5hiwfefbeu

Polynomial-Time Approximation Schemes for Maximizing Gross Substitutes Utility under Budget Constraints [chapter]

Akiyoshi Shioura
2011 Lecture Notes in Computer Science  
More generally, we present a PTAS for maximizing a discrete concave function called an M \ -concave function under budget constraints.  ...  We also consider the maximization of the sum of two M \ -concave functions under a single budget constraint.  ...  Every laminar concave function is a GS utility function. u t Example 3.  ... 
doi:10.1007/978-3-642-23719-5_1 fatcat:2ovtfooiwzfunn2vztbwhsefqm

Polynomial-Time Approximation Schemes for Maximizing Gross Substitutes Utility Under Budget Constraints

Akiyoshi Shioura
2015 Mathematics of Operations Research  
More generally, we present a PTAS for maximizing a discrete concave function called an M \ -concave function under budget constraints.  ...  We also consider the maximization of the sum of two M \ -concave functions under a single budget constraint.  ...  Every laminar concave function is a GS utility function. u t Example 3.  ... 
doi:10.1287/moor.2014.0668 fatcat:rcawxwvdxzfyfmswpok6yvtk6i
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