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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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== Continuous extensions of submodular set functions ==
Often, given a submodular set function that describes the values of various sets, we need to compute the values of ''fractional'' sets. For example: we know that the value of receiving house A and house B is V, and we want to know the value of receiving 40% of house A and 60% of house B. To this end, we need a ''continuous extension'' of the submodular set function.
Formally, a set function <math>f:2^{\Omega}\rightarrow \mathbb{R}</math> with <math>|\Omega|=n</math> can be represented as a function on <math>\{0, 1\}^{n}</math>, by associating each <math>S\subseteq \Omega</math> with a binary vector <math>x^{S}\in \{0, 1\}^{n}</math> such that <math>x_{i}^{S}=1</math> when <math>i\in S</math>, and <math>x_{i}^{S}=0</math> otherwise. A ''continuous [[
Several kinds of continuous extensions of submodular functions are commonly used, which are described below.
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This extension is named after mathematician [[László Lovász]].<ref name="L" /> Consider any vector <math>\mathbf{x}=\{x_1,x_2,\dots,x_n\}</math> such that each <math>0\leq x_i\leq 1</math>. Then the Lovász extension is defined as
<math>f^L(\mathbf{x})=\mathbb{E}(f(\{i|x_i\geq \lambda\}))</math>
where the expectation is over <math>\lambda</math> chosen from the [[uniform distribution (continuous)|uniform distribution]] on the interval <math>[0,1]</math>. The Lovász extension is a convex function if and only if <math>f</math> is a submodular function.
=== 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 book |last=Vondrak |first=Jan |title=Proceedings of the fortieth annual ACM symposium on Theory of computing |chapter=Optimal approximation for the submodular welfare problem in the value oracle model |date=2008-05-17 |chapter-url=https://doi.org/10.1145/1374376.1374389 |series=STOC '08 |___location=New York, NY, USA |publisher=Association for Computing Machinery |pages=67–74 |doi=10.1145/1374376.1374389 |isbn=978-1-60558-047-0|s2cid=170510 }}</ref><ref>{{Cite journal |last1=Calinescu |first1=Gruia |last2=Chekuri |first2=Chandra |last3=Pál |first3=Martin |last4=Vondrák |first4=Jan |date=January 2011 |title=Maximizing a Monotone Submodular Function Subject to a Matroid Constraint |url=http://epubs.siam.org/doi/10.1137/080733991 |journal=SIAM Journal on Computing |language=en |volume=40 |issue=6 |pages=1740–1766 |doi=10.1137/080733991 |issn=0097-5397|url-access=subscription }}</ref><math>F(\mathbf{x})=\sum_{S\subseteq \Omega} f(S) \prod_{i\in S} x_i \prod_{i\notin S} (1-x_i)</math>.
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.
=== Convex closure ===
Consider any vector <math>\mathbf{x}=\{x_1,x_2,\dots,x_n\}</math> such that each <math>0\leq x_i\leq 1</math>. Then the convex closure is defined as <math>f^-(\mathbf{x})=\min\left(\sum_S \alpha_S f(S):\sum_S \alpha_S 1_S=\mathbf{x},\sum_S \alpha_S=1,\alpha_S\geq 0\right)</math>.
The convex closure of any set function is convex over <math>[0,1]^n</math>.
=== Concave closure ===
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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.
=== Submodular set function minimization===
The hardness of minimizing a submodular set function depends on constraints imposed on the problem.
# The unconstrained problem of minimizing a submodular function is computable in [[polynomial time]],<ref name="GLS" /><ref name="Cunningham" /> and even in [[Strongly polynomial|strongly-polynomial]] time.<ref name="IFF" /><ref name="Schrijver" /> Computing the [[minimum cut]] in a graph is a special case of this minimization problem.
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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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== Applications ==
Submodular functions naturally occur in several real world applications, in [[economics]], [[game theory]], [[machine learning]] and [[computer vision]].<ref name="KG" /><ref name="JB" /> Owing to the diminishing returns property, submodular functions naturally model costs of items, since there is often a larger discount, with an increase in the items one buys. Submodular functions model notions of complexity, similarity and cooperation when they appear in minimization problems. In maximization problems, on the other hand, they model notions of diversity, information and coverage.
== See also ==
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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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