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{{Short description|Set-to-real map with diminishing returns}}
{{Use American English|date = January 2019}}
In mathematics, a '''submodular set function''' (also known as a '''submodular function''') is a [[set function]] that, informally, describes the relationship between a set of inputs and an output, where adding more of one input has a decreasing additional benefit ([[diminishing returns]]). The natural [[diminishing returns]] property which makes them suitable for many applications, including [[approximation algorithms]], [[game theory]] (as functions modeling user preferences) and [[electrical network]]s. Recently, submodular functions have also found
== Definition ==
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=== Non-monotone ===
A submodular function that is not monotone is called ''non-monotone''. In particular, a function is called non-monotone if it has the property that adding more elements to a set can decrease the value of the function. More formally, the function <math> f </math> is non-monotone if there are sets <math>S,T</math> in its ___domain s.t. <math> S \subset T </math> and <math>f(S)> f(T)</math>.
==== Symmetric ====
A non-monotone submodular function <math>f</math> is called ''symmetric'' if for every <math>S\subseteq \Omega</math> we have that <math>f(S)=f(\Omega-S)</math>.
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=== Multilinear extension ===
Consider any vector <math>\mathbf{x}=\{x_1,x_2,\ldots,x_n\}</math> such that each <math>0\leq x_i\leq 1</math>. Then the multilinear extension is defined as <ref>{{Cite
Intuitively, ''x<sub>i</sub>'' represents the probability that item ''i'' is chosen for the set. For every set ''S'', the two inner products represent the probability that the chosen set is exactly ''S''. Therefore, the sum represents the expected value of ''f'' for the set formed by choosing each item ''i'' at random with probability xi, independently of the other items.
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# Consider a random process where a set <math>T</math> is chosen with each element in <math>\Omega</math> being included in <math>T</math> independently with probability <math>p</math>. Then the following inequality is true <math>\mathbb{E}[f(T)]\geq p f(\Omega)+(1-p) f(\varnothing)</math> where <math>\varnothing</math> is the empty set. More generally consider the following random process where a set <math>S</math> is constructed as follows. For each of <math>1\leq i\leq l, A_i\subseteq \Omega</math> construct <math>S_i</math> by including each element in <math>A_i</math> independently into <math>S_i</math> with probability <math>p_i</math>. Furthermore let <math>S=\cup_{i=1}^l S_i</math>. Then the following inequality is true <math>\mathbb{E}[f(S)]\geq \sum_{R\subseteq [l]} \Pi_{i\in R}p_i \Pi_{i\notin R}(1-p_i)f(\cup_{i\in R}A_i)</math>.{{Citation needed|date=November 2013}}
== Optimization problems{{Anchor|optimization}} ==
Submodular functions have properties which are very similar to [[convex function|convex]] and [[concave function]]s. For this reason, an [[optimization problem]] which concerns optimizing a convex or concave function can also be described as the problem of maximizing or minimizing a submodular function subject to some constraints.
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# The problem of maximizing a non-negative submodular function admits a 1/2 approximation algorithm.<ref name="FMV" /><ref name="BFNS" /> Computing the [[maximum cut]] of a graph is a special case of this problem.
# The problem of maximizing a monotone submodular function subject to a cardinality constraint admits a <math>1 - 1/e</math> approximation algorithm.<ref name="NVF" />
# The problem of maximizing a monotone submodular function subject to a [[matroid]] constraint (which subsumes the case above) also admits a <math>1 - 1/e</math> approximation algorithm.<ref name="CCPV" /><ref name="FNS" /><ref name="FW" />
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<ref name="IFF">{{cite journal |first1=S. |last1=Iwata |first2=L. |last2=Fleischer |first3=S. |last3=Fujishige |title=A combinatorial strongly polynomial algorithm for minimizing submodular functions |journal=J. ACM |volume=48 |year=2001 |issue=4 |pages=761–777 |doi=10.1145/502090.502096 |s2cid=888513 }}</ref>
<ref name="Schrijver">{{cite journal |author-link=Alexander Schrijver |first=A. |last=Schrijver |title=A combinatorial algorithm minimizing submodular functions in strongly polynomial time |journal=J. Combin. Theory Ser. B |volume=80 |year=2000 |issue=2 |pages=346–355 |doi=10.1006/jctb.2000.1989 |url=https://ir.cwi.nl/pub/2108 |doi-access=free }}</ref>
<ref name="IJB">R. Iyer, [[Stefanie Jegelka|S. Jegelka]] and J. Bilmes, Fast Semidifferential based submodular function optimization, Proc. ICML (2013).</ref>
<ref name="IB">R. Iyer and J. Bilmes, Submodular Optimization Subject to Submodular Cover and Submodular Knapsack Constraints, In Advances of NIPS (2013).</ref>
<ref name="IBUAI">R. Iyer and J. Bilmes, Algorithms for Approximate Minimization of the Difference between Submodular Functions, In Proc. UAI (2012).</ref>
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<ref name="KG1">A. Krause and C. Guestrin, Near-optimal nonmyopic value of information in graphical models, UAI-2005.</ref>
<ref name="FNS">M. Feldman, J. Naor and R. Schwartz, A unified continuous greedy algorithm for submodular maximization, Proc. of 52nd FOCS (2011).</ref>
<ref name="L">{{cite
<ref name="BF">{{cite encyclopedia |last1=Buchbinder |first1=Niv |last2=Feldman |first2=Moran |title=Submodular Functions Maximization Problems |encyclopedia= Handbook of Approximation Algorithms and Metaheuristics, Second Edition: Methodologies and Traditional Applications |year=2018 |editor1-last=Gonzalez |editor1-first=Teofilo F. |publisher=Chapman and Hall/CRC |doi=10.1201/9781351236423 |isbn=9781351236423 |url=https://www.taylorfrancis.com/chapters/edit/10.1201/9781351236423-42/submodular-functions-maximization-problems-niv-buchbinder-moran-feldman|url-access=subscription }}</ref>
<ref name="JV2">{{Cite web|last=Vondrák|first=Jan|title=Polyhedral techniques in combinatorial optimization: Lecture 17|url=https://theory.stanford.edu/~jvondrak/CS369P/lec17.pdf}}</ref>
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*{{Citation|last=Lee|first=Jon|author-link=Jon Lee (mathematician)|year= 2004 |title=A First Course in Combinatorial Optimization |publisher=[[Cambridge University Press]]|isbn= 0-521-01012-8}}
*{{Citation|last=Fujishige|first=Satoru|year=2005|title=Submodular Functions and Optimization|publisher=[[Elsevier]]|isbn=0-444-52086-4}}
*{{Citation|last=Narayanan|first=H.|year= 1997 |title=Submodular Functions and Electrical Networks|publisher=Elsevier |isbn= 0-444-82523-1}}
*{{citation | last=Oxley | first=James G. | title=Matroid theory | series=Oxford Science Publications | ___location=Oxford | publisher=[[Oxford University Press]] | year=1992 | isbn=0-19-853563-5 | zbl=0784.05002 }}
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