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Minimizing a monotone concave function with laminar covering constraints

2008
*
Discrete Applied Mathematics
*

Given

doi:10.1016/j.dam.2007.04.016
fatcat:cxsflulerrdvfamvqvrvhlifcq
*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. ...##
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Minimizing a Monotone Concave Function with Laminar Covering Constraints
[chapter]

2005
*
Lecture Notes in Computer Science
*

Given

doi:10.1007/11602613_9
fatcat:gsnejuc4brg6da7lpmzffufgg4
*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 ...##
###
Contents

2008
*
Discrete Applied Mathematics
*

Fujishige

doi:10.1016/s0166-218x(08)00287-4
fatcat:n7ujjgi24bey7ieuwq6lquodsa
*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. ...##
###
Deep Submodular Functions: Definitions and Learning

2016
*
Neural Information Processing Systems
*

We define DSFs and situate them within the broader context of classes of submodular

dblp:conf/nips/DolhanskyB16
fatcat:t3zsaouy2naenc4xuseitdz7xa
*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 . ...##
###
The submodular joint replenishment problem

2015
*
Mathematical programming
*

In this paper, the cost of an order, also known as

doi:10.1007/s10107-015-0920-3
fatcat:gjtmzwdpl5c6hfnjszac63tgz4
*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 ...##
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On Multiplicative Weight Updates for Concave and Submodular Function Maximization

2015
*
Proceedings of the 2015 Conference on Innovations in Theoretical Computer Science - ITCS '15
*

The framework allows for

doi:10.1145/2688073.2688086
dblp:conf/innovations/ChekuriJV15
fatcat:qh5rdgdmtnbu7c6ataszrho3eq
*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*. ...##
###
Deep Submodular Functions
[article]

2017
*
arXiv
*
pre-print

We start

arXiv:1701.08939v1
fatcat:zvbpgq7d2feqrantvl7hi34jum
*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)). ...##
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Polyhedral aspects of Submodularity, Convexity and Concavity
[article]

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 ...

##
###
Provable Variational Inference for Constrained Log-Submodular Models

2018
*
Neural Information Processing Systems
*

In addition to providing completely tractable and well-understood variational approximations, our approach results in the

dblp:conf/nips/DjolongaJ018
fatcat:ag4xpvgvzrh4jnx5o4xckcoyc4
*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*. ...##
###
Minimum Cost Source Location Problems with Flow Requirements
[chapter]

2006
*
Lecture Notes in Computer Science
*

κ), where the source location problems are to find

doi:10.1007/11682462_70
fatcat:ik3t7xmsjveadbnxdge5lsgqfy
*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 ...##
###
Submodular Functions: Optimization and Approximation

2011
*
Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)
*

For submodular

doi:10.1142/9789814324359_0173
fatcat:uspd2ma2uzco3kvzqiytgxz45a
*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*. ...##
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Multiwinner Elections with Diversity Constraints
[article]

2017
*
arXiv
*
pre-print

We develop

arXiv:1711.06527v2
fatcat:bjl2nvzvjbbshpv6mhatr4llve
*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 ). ...##
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Discrete convex analysis: A tool for economics and game theory

2016
*
Journal of Mechanism and Institution Design
*

Recently, it has been recognized as

doi:10.22574/jmid.2016.12.005
fatcat:e73iapgazbcxljnr5hiwfefbeu
*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. ...##
###
Polynomial-Time Approximation Schemes for Maximizing Gross Substitutes Utility under Budget Constraints
[chapter]

2011
*
Lecture Notes in Computer Science
*

More generally, we present

doi:10.1007/978-3-642-23719-5_1
fatcat:2ovtfooiwzfunn2vztbwhsefqm
*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. ...##
###
Polynomial-Time Approximation Schemes for Maximizing Gross Substitutes Utility Under Budget Constraints

2015
*
Mathematics of Operations Research
*

More generally, we present

doi:10.1287/moor.2014.0668
fatcat:rcawxwvdxzfyfmswpok6yvtk6i
*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. ...
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