Submodular set function: Difference between revisions

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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>.

A submodular function that is not monotone is called ''non-monotone''.

==== 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 journalbook |last=Vondrak |first=Jan |datetitle=2008-05-17Proceedings of the fortieth annual ACM symposium on Theory of computing |titlechapter=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 |journal=Proceedings of the fortieth annual ACM symposium on Theory of computing |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 |lastlast1=Calinescu |firstfirst1=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_i'' 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" />{{page needed|date=October 2020}}<ref>{{Cite web|last=Williamson|first=David P.|title=Bridging Continuous and Discrete Optimization: Lecture 23|url=https://people.orie.cornell.edu/dpw/orie6334/lecture23.pdf}}</ref> The [[maximum coverage problem]] is a special case of this problem.

# 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 journalbook |author-link1=László Lovász |last1=Lovász |first1=L. |datetitle=1983Mathematical Programming the State of the Art |titlechapter=Submodular functions and convexity |urldate=1983 |journalchapter-url=Mathematical Programming the State of the Art |pages=235–257 |doi=10.1007/978-3-642-68874-4_10 |isbn=978-3-642-68876-8 |s2cid=117358746 }}</ref>

<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 }}