ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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<title>Submodular set function</title>

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Submodular set function

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In mathematics, a submodular set function (also known as a submodular function) is a <a href="Set_function" title="Set function">set function</a> that, informally, describes the relationship between a set of inputs and an output, where adding more of one input has a decreasing additional benefit (<a href="Diminishing_returns" title="Diminishing returns">diminishing returns</a>). The natural <a href="Diminishing_returns" title="Diminishing returns">diminishing returns</a> property which makes them suitable for many applications, including <a href="Approximation_algorithms" class="mw-redirect" title="Approximation algorithms">approximation algorithms</a>, <a href="Game_theory" title="Game theory">game theory</a> (as functions modeling user preferences) and <a href="Electrical_network" title="Electrical network">electrical networks</a>. Recently, submodular functions have also found utility in several real world problems in <a href="Machine_learning" title="Machine learning">machine learning</a> and <a href="Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>, including <a href="Automatic_summarization" title="Automatic summarization">automatic summarization</a>, <a href="Multi-document_summarization" title="Multi-document summarization">multi-document summarization</a>, <a href="Feature_selection" title="Feature selection">feature selection</a>, <a href="Active_learning_(machine_learning)" title="Active learning (machine learning)">active learning</a>, sensor placement, image collection summarization and many other domains.<a href="#cite_note-LB-1">[1]</a><a href="#cite_note-TIWB-2">[2]</a><a href="#cite_note-KG1-3">[3]</a><a href="#cite_note-KG-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

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</math><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"> is a finite <a href="Set_(mathematics)" title="Set (mathematics)">set</a>, a submodular function is a set function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:2^{\Omega }\rightarrow \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle f:2^{\Omega }\rightarrow \mathbb {R} }</annotation>

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</math><img src="./3a7218c70cf496cabc1e01ea1bb5037a29213ac0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.089ex; height:3.009ex;" alt="{\displaystyle f:2^{\Omega }\rightarrow \mathbb {R} }" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\Omega }}">

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<annotation encoding="application/x-tex">{\displaystyle 2^{\Omega }}</annotation>

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</math><img src="./56c53a10995552345e60c76e601ae3afdbd49158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.581ex; height:2.676ex;" alt="{\displaystyle 2^{\Omega }}" loading="lazy"> denotes the <a href="Power_set#Representing_subsets_as_functions" title="Power set">power set</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

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</math><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy">, which satisfies one of the following equivalent conditions.<a href="#cite_note-5">[5]</a>

<ol><li>For every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y\subseteq \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle X,Y\subseteq \Omega }</annotation>

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</math><img src="./5d47da32f1d39807e1ad4e737cd76bebb2683859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.564ex; height:2.509ex;" alt="{\displaystyle X,Y\subseteq \Omega }" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subseteq Y}">

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<annotation encoding="application/x-tex">{\displaystyle X\subseteq Y}</annotation>

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</math><img src="./894a17f6bbae8bb911fb08785e051b353e832ea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.852ex; height:2.343ex;" alt="{\displaystyle X\subseteq Y}" loading="lazy"> and every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \Omega \setminus Y}">

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<annotation encoding="application/x-tex">{\displaystyle x\in \Omega \setminus Y}</annotation>

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</math><img src="./adb7cafa56041710f1fb3c4a06fc3733164ca440.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.816ex; height:2.843ex;" alt="{\displaystyle x\in \Omega \setminus Y}" loading="lazy"> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X\cup \{x\})-f(X)\geq f(Y\cup \{x\})-f(Y)}">

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<annotation encoding="application/x-tex">{\displaystyle f(X\cup \{x\})-f(X)\geq f(Y\cup \{x\})-f(Y)}</annotation>

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</math><img src="./bc0a71503999ef3b550e69904aa881420f6b011c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.112ex; height:2.843ex;" alt="{\displaystyle f(X\cup \{x\})-f(X)\geq f(Y\cup \{x\})-f(Y)}" loading="lazy">.</li>

<li>For every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S,T\subseteq \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle S,T\subseteq \Omega }</annotation>

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</math><img src="./c0d257a9a2eb787c5690a306ab0cdccba7c2893b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.946ex; height:2.509ex;" alt="{\displaystyle S,T\subseteq \Omega }" loading="lazy"> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)+f(T)\geq f(S\cup T)+f(S\cap T)}">

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<annotation encoding="application/x-tex">{\displaystyle f(S)+f(T)\geq f(S\cup T)+f(S\cap T)}</annotation>

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</math><img src="./f21068db00908ee3727f8f8f64990b2a68ea7c09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.702ex; height:2.843ex;" alt="{\displaystyle f(S)+f(T)\geq f(S\cup T)+f(S\cap T)}" loading="lazy">.</li>

<li>For every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subseteq \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle X\subseteq \Omega }</annotation>

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</math><img src="./8d9008c76549e7c46477a26022fa1e4ca10741e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.757ex; height:2.343ex;" alt="{\displaystyle X\subseteq \Omega }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},x_{2}\in \Omega \backslash X}">

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<annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2}\in \Omega \backslash X}</annotation>

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</math><img src="./b708176cf6476b287cd7437a9b62a7b0542090b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.463ex; height:2.843ex;" alt="{\displaystyle x_{1},x_{2}\in \Omega \backslash X}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}\neq x_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{1}\neq x_{2}}</annotation>

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</math><img src="./ae809ae1f500b6f959ee08a5895ed2545ceb6827.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.866ex; height:2.676ex;" alt="{\displaystyle x_{1}\neq x_{2}}" loading="lazy"> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X\cup \{x_{1}\})+f(X\cup \{x_{2}\})\geq f(X\cup \{x_{1},x_{2}\})+f(X)}">

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<annotation encoding="application/x-tex">{\displaystyle f(X\cup \{x_{1}\})+f(X\cup \{x_{2}\})\geq f(X\cup \{x_{1},x_{2}\})+f(X)}</annotation>

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</math><img src="./3da73513b2d772e935cd50b6919beabd00c21f30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.343ex; height:2.843ex;" alt="{\displaystyle f(X\cup \{x_{1}\})+f(X\cup \{x_{2}\})\geq f(X\cup \{x_{1},x_{2}\})+f(X)}" loading="lazy">.</li></ol>

A nonnegative submodular function is also a <a href="Subadditive_set_function" title="Subadditive set function">subadditive</a> function, but a subadditive function need not be submodular.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

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</math><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"> is not assumed finite, then the above conditions are not equivalent. In particular a function

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)=1}">

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<annotation encoding="application/x-tex">{\displaystyle f(S)=1}</annotation>

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</math><img src="./edec56e6744701f258f11db5a082224b781d9c00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.848ex; height:2.843ex;" alt="{\displaystyle f(S)=1}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is finite and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)=0}">

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<annotation encoding="application/x-tex">{\displaystyle f(S)=0}</annotation>

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</math><img src="./1936b62454ff88f161f986bc8683eb2b9366eecd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.848ex; height:2.843ex;" alt="{\displaystyle f(S)=0}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is infinite

satisfies the first condition above, but the second condition fails when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> are infinite sets with finite intersection.

<div class="mw-heading mw-heading2"><h2 id="Types_and_examples_of_submodular_functions">Types and examples of submodular functions</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Monotone">Monotone</h3></div>

A set function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is monotone if for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\subseteq S}">

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<annotation encoding="application/x-tex">{\displaystyle T\subseteq S}</annotation>

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</math><img src="./8aa9f13bef3d0e5269f5c3ae3884ad612d64cf92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.234ex; height:2.343ex;" alt="{\displaystyle T\subseteq S}" loading="lazy"> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(T)\leq f(S)}">

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<annotation encoding="application/x-tex">{\displaystyle f(T)\leq f(S)}</annotation>

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</math><img src="./beea7b4d11cafd4dbaac6f3de4aca157b676a070.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.41ex; height:2.843ex;" alt="{\displaystyle f(T)\leq f(S)}" loading="lazy">. Examples of monotone submodular functions include:

<dl><dt>Linear (Modular) functions</dt>

<dd>Any function of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)=\sum _{i\in S}w_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle f(S)=\sum _{i\in S}w_{i}}</annotation>

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</math><img src="./2100ee57e74c2d4173913a785467b1e5e25c9fc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.891ex; height:5.676ex;" alt="{\displaystyle f(S)=\sum _{i\in S}w_{i}}" loading="lazy"> is called a linear function. Additionally if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall i,w_{i}\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle \forall i,w_{i}\geq 0}</annotation>

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</math><img src="./67cf41cbc2faefd76cc644e6e98182dc55d55668.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.854ex; height:2.509ex;" alt="{\displaystyle \forall i,w_{i}\geq 0}" loading="lazy"> then f is monotone.</dd>

<dt><a href="Budget-additive_valuation" title="Budget-additive valuation">Budget-additive functions</a></dt>

<dd>Any function of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)=\min \left\{B,~\sum _{i\in S}w_{i}\right\}}">

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<annotation encoding="application/x-tex">{\displaystyle f(S)=\min \left\{B,~\sum _{i\in S}w_{i}\right\}}</annotation>

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</math><img src="./248b50a6d6479d0dce69c972d22d69b8ea03d759.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.666ex; height:7.509ex;" alt="{\displaystyle f(S)=\min \left\{B,~\sum _{i\in S}w_{i}\right\}}" loading="lazy"> for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle w_{i}\geq 0}</annotation>

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</math><img src="./5bc4e26452494d1471216374f94480d6cd17173e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.725ex; height:2.509ex;" alt="{\displaystyle w_{i}\geq 0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle B\geq 0}</annotation>

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</math><img src="./db83a396bf5ff2d5b36198835c75b7b74b4f3ace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.025ex; height:2.343ex;" alt="{\displaystyle B\geq 0}" loading="lazy"> is called budget additive.<a href="#cite_note-BF-6">[6]</a></dd>

<dt>Coverage functions</dt>

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<annotation encoding="application/x-tex">{\displaystyle \Omega =\{E_{1},E_{2},\ldots ,E_{n}\}}</annotation>

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</math><img src="./61a95c84e60864d8e6aa68d7121740367d501dc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.786ex; height:2.843ex;" alt="{\displaystyle \Omega =\{E_{1},E_{2},\ldots ,E_{n}\}}" loading="lazy"> be a collection of subsets of some <a href="Matroid" title="Matroid">ground set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega '}">

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<annotation encoding="application/x-tex">{\displaystyle \Omega '}</annotation>

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</math><img src="./e7816f1c1988daecea948e9ea48170b028d97602.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.363ex; height:2.509ex;" alt="{\displaystyle \Omega '}" loading="lazy">. The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)=\left|\bigcup _{E_{i}\in S}E_{i}\right|}">

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<annotation encoding="application/x-tex">{\displaystyle f(S)=\left|\bigcup _{E_{i}\in S}E_{i}\right|}</annotation>

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</math><img src="./a20d2e3a4cf7efa30dcbaa37365fde3509af6117.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.875ex; height:7.843ex;" alt="{\displaystyle f(S)=\left|\bigcup _{E_{i}\in S}E_{i}\right|}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq \Omega }">

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<annotation encoding="application/x-tex">{\displaystyle S\subseteq \Omega }</annotation>

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</math><img src="./926e3bc2815ca680f0a752381de3770f44d16b5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.276ex; height:2.343ex;" alt="{\displaystyle S\subseteq \Omega }" loading="lazy"> is called a coverage function. This can be generalized by adding non-negative weights to the elements.</dd>

<dt><a href="Entropy_(information_theory)" title="Entropy (information theory)">Entropy</a></dt>

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<annotation encoding="application/x-tex">{\displaystyle \Omega =\{X_{1},X_{2},\ldots ,X_{n}\}}</annotation>

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</math><img src="./33a6f158cbc712e37da1a286c454a8eb91a26f6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.413ex; height:2.843ex;" alt="{\displaystyle \Omega =\{X_{1},X_{2},\ldots ,X_{n}\}}" loading="lazy"> be a set of <a href="Random_variables" class="mw-redirect" title="Random variables">random variables</a>. Then for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\subseteq \Omega }</annotation>

</semantics>

</math><img src="./926e3bc2815ca680f0a752381de3770f44d16b5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.276ex; height:2.343ex;" alt="{\displaystyle S\subseteq \Omega }" loading="lazy"> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(S)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(S)}</annotation>

</semantics>

</math><img src="./1061a0762c27da11a9f4a68b4775d3273f6c9d2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.372ex; height:2.843ex;" alt="{\displaystyle H(S)}" loading="lazy"> is a submodular function, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(S)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H(S)}</annotation>

</semantics>

</math><img src="./1061a0762c27da11a9f4a68b4775d3273f6c9d2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.372ex; height:2.843ex;" alt="{\displaystyle H(S)}" loading="lazy"> is the entropy of the set of random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">, a fact known as <a href="Entropic_vector#Shannon-type_inequalities_and_Γn" title="Entropic vector">Shannon's inequality</a>.<a href="#cite_note-7">[7]</a> Further inequalities for the entropy function are known to hold, see <a href="Entropic_vector" title="Entropic vector">entropic vector</a>.</dd>

<dt><a href="Matroid" title="Matroid">Matroid</a> <a href="Matroid_rank" title="Matroid rank">rank functions</a></dt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega =\{e_{1},e_{2},\dots ,e_{n}\}}</annotation>

</semantics>

</math><img src="./5768c420a952b7ccccd19c3acf3bc3ca247ecb15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.891ex; height:2.843ex;" alt="{\displaystyle \Omega =\{e_{1},e_{2},\dots ,e_{n}\}}" loading="lazy"> be the ground set on which a matroid is defined. Then the rank function of the matroid is a submodular function.<a href="#cite_note-F22-8">[8]</a></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Non-monotone">Non-monotone</h3></div>

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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is non-monotone if there are sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S,T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S,T}</annotation>

</semantics>

</math><img src="./4972a806d8142ce359a93ddb4c99340d62ef774e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.169ex; height:2.509ex;" alt="{\displaystyle S,T}" loading="lazy"> in its domain s.t. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subset T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\subset T}</annotation>

</semantics>

</math><img src="./e9bf7ab27d85f6a7220e58f2fee59077aaefc36f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.234ex; height:2.176ex;" alt="{\displaystyle S\subset T}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)>f(T)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(S)>f(T)}</annotation>

</semantics>

</math><img src="./dcbb27665920e3a5176eba19d40f5565ab3042d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.41ex; height:2.843ex;" alt="{\displaystyle f(S)>f(T)}" loading="lazy">.

<div class="mw-heading mw-heading4"><h4 id="Symmetric">Symmetric</h4></div>

A non-monotone submodular function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is called symmetric if for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\subseteq \Omega }</annotation>

</semantics>

</math><img src="./926e3bc2815ca680f0a752381de3770f44d16b5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.276ex; height:2.343ex;" alt="{\displaystyle S\subseteq \Omega }" loading="lazy"> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)=f(\Omega -S)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(S)=f(\Omega -S)}</annotation>

</semantics>

</math><img src="./9f114d9780832df7171e3c2cf02557029ef53378.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.791ex; height:2.843ex;" alt="{\displaystyle f(S)=f(\Omega -S)}" loading="lazy">.

Examples of symmetric non-monotone submodular functions include:

<dl><dt>Graph cuts</dt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega =\{v_{1},v_{2},\dots ,v_{n}\}}</annotation>

</semantics>

</math><img src="./db3b6195ac98af44548d8449bdd22daccc6a8c03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.023ex; height:2.843ex;" alt="{\displaystyle \Omega =\{v_{1},v_{2},\dots ,v_{n}\}}" loading="lazy"> be the vertices of a <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a>. For any set of vertices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\subseteq \Omega }</annotation>

</semantics>

</math><img src="./926e3bc2815ca680f0a752381de3770f44d16b5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.276ex; height:2.343ex;" alt="{\displaystyle S\subseteq \Omega }" loading="lazy"> let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(S)}</annotation>

</semantics>

</math><img src="./99a865d36b32ea1d60f15dc3093f5b28093f192b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.587ex; height:2.843ex;" alt="{\displaystyle f(S)}" loading="lazy"> denote the number of edges <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=(u,v)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e=(u,v)}</annotation>

</semantics>

</math><img src="./630599cea0ef5d4484bd057fb50334cf3fd9433c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.482ex; height:2.843ex;" alt="{\displaystyle e=(u,v)}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\in S}</annotation>

</semantics>

</math><img src="./a81bd6308fa73ba1bec537a7b5f20429936b0fa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.67ex; height:2.176ex;" alt="{\displaystyle u\in S}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in \Omega -S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\in \Omega -S}</annotation>

</semantics>

</math><img src="./362002b18c49b305f72366a90704f1866e21a3a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.986ex; height:2.343ex;" alt="{\displaystyle v\in \Omega -S}" loading="lazy">. This can be generalized by adding non-negative weights to the edges.</dd>

<dt><a href="Mutual_information" title="Mutual information">Mutual information</a></dt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega =\{X_{1},X_{2},\ldots ,X_{n}\}}</annotation>

</semantics>

</math><img src="./33a6f158cbc712e37da1a286c454a8eb91a26f6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.413ex; height:2.843ex;" alt="{\displaystyle \Omega =\{X_{1},X_{2},\ldots ,X_{n}\}}" loading="lazy"> be a set of <a href="Random_variable" title="Random variable">random variables</a>. Then for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\subseteq \Omega }</annotation>

</semantics>

</math><img src="./926e3bc2815ca680f0a752381de3770f44d16b5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.276ex; height:2.343ex;" alt="{\displaystyle S\subseteq \Omega }" loading="lazy"> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)=I(S;\Omega -S)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(S)=I(S;\Omega -S)}</annotation>

</semantics>

</math><img src="./de107d78b086cbe50a4f9ca1d44395c69225503e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.217ex; height:2.843ex;" alt="{\displaystyle f(S)=I(S;\Omega -S)}" loading="lazy"> is a submodular function, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(S;\Omega -S)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I(S;\Omega -S)}</annotation>

</semantics>

</math><img src="./14942e1269dcaee4c6aa1fb7fceb55c3992a8327.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.532ex; height:2.843ex;" alt="{\displaystyle I(S;\Omega -S)}" loading="lazy"> is the mutual information.</dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Asymmetric">Asymmetric</h4></div>

A non-monotone submodular function which is not symmetric is called asymmetric.

<dl><dt>Directed cuts</dt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega =\{v_{1},v_{2},\dots ,v_{n}\}}</annotation>

</semantics>

</math><img src="./db3b6195ac98af44548d8449bdd22daccc6a8c03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.023ex; height:2.843ex;" alt="{\displaystyle \Omega =\{v_{1},v_{2},\dots ,v_{n}\}}" loading="lazy"> be the vertices of a <a href="Directed_graph" title="Directed graph">directed graph</a>. For any set of vertices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\subseteq \Omega }</annotation>

</semantics>

</math><img src="./926e3bc2815ca680f0a752381de3770f44d16b5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.276ex; height:2.343ex;" alt="{\displaystyle S\subseteq \Omega }" loading="lazy"> let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(S)}</annotation>

</semantics>

</math><img src="./99a865d36b32ea1d60f15dc3093f5b28093f192b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.587ex; height:2.843ex;" alt="{\displaystyle f(S)}" loading="lazy"> denote the number of edges <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=(u,v)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e=(u,v)}</annotation>

</semantics>

</math><img src="./630599cea0ef5d4484bd057fb50334cf3fd9433c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.482ex; height:2.843ex;" alt="{\displaystyle e=(u,v)}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\in S}</annotation>

</semantics>

</math><img src="./a81bd6308fa73ba1bec537a7b5f20429936b0fa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.67ex; height:2.176ex;" alt="{\displaystyle u\in S}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\in \Omega -S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\in \Omega -S}</annotation>

</semantics>

</math><img src="./362002b18c49b305f72366a90704f1866e21a3a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.986ex; height:2.343ex;" alt="{\displaystyle v\in \Omega -S}" loading="lazy">. This can be generalized by adding non-negative weights to the directed edges.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Continuous_extensions_of_submodular_set_functions">Continuous extensions of submodular set functions</h2></div>

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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:2^{\Omega }\rightarrow \mathbb {R} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:2^{\Omega }\rightarrow \mathbb {R} }</annotation>

</semantics>

</math><img src="./3a7218c70cf496cabc1e01ea1bb5037a29213ac0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.089ex; height:3.009ex;" alt="{\displaystyle f:2^{\Omega }\rightarrow \mathbb {R} }" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Omega |=n}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\Omega |=n}</annotation>

</semantics>

</math><img src="./bb9efe17366259565cda7c44810a0ba710d9d85f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.465ex; height:2.843ex;" alt="{\displaystyle |\Omega |=n}" loading="lazy"> can be represented as a function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1\}^{n}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{0,1\}^{n}}</annotation>

</semantics>

</math><img src="./3cc07e486d73e18382d0d8d205149f0923ed0586.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.902ex; height:2.843ex;" alt="{\displaystyle \{0,1\}^{n}}" loading="lazy">, by associating each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\subseteq \Omega }</annotation>

</semantics>

</math><img src="./926e3bc2815ca680f0a752381de3770f44d16b5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.276ex; height:2.343ex;" alt="{\displaystyle S\subseteq \Omega }" loading="lazy"> with a binary vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{S}\in \{0,1\}^{n}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{S}\in \{0,1\}^{n}}</annotation>

</semantics>

</math><img src="./5fa63c63b3c796c1232a6f2ac080434eba2bc95d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.365ex; height:3.176ex;" alt="{\displaystyle x^{S}\in \{0,1\}^{n}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}^{S}=1}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}^{S}=1}</annotation>

</semantics>

</math><img src="./1878ad9a31ed8976f08db82668441e1f824c0c5f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.883ex; height:3.176ex;" alt="{\displaystyle x_{i}^{S}=1}" loading="lazy"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i\in S}</annotation>

</semantics>

</math><img src="./939944399fbd10a8440736e746193d90a6725781.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.142ex; height:2.176ex;" alt="{\displaystyle i\in S}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}^{S}=0}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}^{S}=0}</annotation>

</semantics>

</math><img src="./f641568f5ab1be4076173ecc4dd58d4c39834fbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.883ex; height:3.176ex;" alt="{\displaystyle x_{i}^{S}=0}" loading="lazy"> otherwise. A continuous <a href="Restriction_(mathematics)#Extension_of_a_function" title="Restriction (mathematics)">extension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is a continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:[0,1]^{n}\rightarrow \mathbb {R} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:[0,1]^{n}\rightarrow \mathbb {R} }</annotation>

</semantics>

</math><img src="./1ba21d0641b79959c20e404898baa9d33bb0b546.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.841ex; height:2.843ex;" alt="{\displaystyle F:[0,1]^{n}\rightarrow \mathbb {R} }" loading="lazy">, that matches the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \{0,1\}^{n}}">

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</math><img src="./a2fe5fa2f2f1720e40e59457a01bb8bf14e0e4fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.073ex; height:2.843ex;" alt="{\displaystyle x\in \{0,1\}^{n}}" loading="lazy">, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x^{S})=f(S)}">

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<annotation encoding="application/x-tex">{\displaystyle F(x^{S})=f(S)}</annotation>

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</math><img src="./a50034b9463f201af6b748498faeef05c0110dcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.858ex; height:3.176ex;" alt="{\displaystyle F(x^{S})=f(S)}" loading="lazy">.

Several kinds of continuous extensions of submodular functions are commonly used, which are described below.

<div class="mw-heading mw-heading3"><h3 id="Lovász_extension">Lovász extension</h3></div>

This extension is named after mathematician <a href="L%C3%A1szl%C3%B3_Lov%C3%A1sz" title="László Lovász">László Lovász</a>.<a href="#cite_note-L-9">[9]</a> Consider any vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}}</annotation>

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</math><img src="./37773dae83e9f47788a37b3c174636c06b89e590.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.362ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}}" loading="lazy"> such that each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq x_{i}\leq 1}">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq x_{i}\leq 1}</annotation>

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</math><img src="./f96edd7ea2da465fc4fffca76a3a6c4532bc1642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.651ex; height:2.509ex;" alt="{\displaystyle 0\leq x_{i}\leq 1}" loading="lazy">. Then the Lovász extension is defined as

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<annotation encoding="application/x-tex">{\displaystyle f^{L}(\mathbf {x} )=\mathbb {E} (f(\{i|x_{i}\geq \lambda \}))}</annotation>

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</math><img src="./d6dcb33c95caf317aa70f08729eed37a6da6d5ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.796ex; height:3.176ex;" alt="{\displaystyle f^{L}(\mathbf {x} )=\mathbb {E} (f(\{i|x_{i}\geq \lambda \}))}" loading="lazy">

where the expectation is over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

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<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

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</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> chosen from the <a href="Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform distribution</a> on the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

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<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

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</math><img src="./738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy">. The Lovász extension is a convex function if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is a submodular function.

<div class="mw-heading mw-heading3"><h3 id="Multilinear_extension">Multilinear extension</h3></div>

Consider any vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =\{x_{1},x_{2},\ldots ,x_{n}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =\{x_{1},x_{2},\ldots ,x_{n}\}}</annotation>

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</math><img src="./5f5a8ab0f63758335b4ffa6c73b68bd4fdcab7aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.362ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} =\{x_{1},x_{2},\ldots ,x_{n}\}}" loading="lazy"> such that each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq x_{i}\leq 1}">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq x_{i}\leq 1}</annotation>

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</math><img src="./f96edd7ea2da465fc4fffca76a3a6c4532bc1642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.651ex; height:2.509ex;" alt="{\displaystyle 0\leq x_{i}\leq 1}" loading="lazy">. Then the multilinear extension is defined as <a href="#cite_note-10">[10]</a><a href="#cite_note-11">[11]</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\mathbf {x} )=\sum _{S\subseteq \Omega }f(S)\prod _{i\in S}x_{i}\prod _{i\notin S}(1-x_{i})}">

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<annotation encoding="application/x-tex">{\displaystyle F(\mathbf {x} )=\sum _{S\subseteq \Omega }f(S)\prod _{i\in S}x_{i}\prod _{i\notin S}(1-x_{i})}</annotation>

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</math><img src="./da99a907a49ee264b964efc61cb9c50fd4c2df04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:33.73ex; height:6.009ex;" alt="{\displaystyle F(\mathbf {x} )=\sum _{S\subseteq \Omega }f(S)\prod _{i\in S}x_{i}\prod _{i\notin S}(1-x_{i})}" loading="lazy">.

Intuitively, xi 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.

<div class="mw-heading mw-heading3"><h3 id="Convex_closure">Convex closure</h3></div>

Consider any vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}}</annotation>

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</math><img src="./37773dae83e9f47788a37b3c174636c06b89e590.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.362ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}}" loading="lazy"> such that each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq x_{i}\leq 1}">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq x_{i}\leq 1}</annotation>

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</math><img src="./f96edd7ea2da465fc4fffca76a3a6c4532bc1642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.651ex; height:2.509ex;" alt="{\displaystyle 0\leq x_{i}\leq 1}" loading="lazy">. Then the convex closure is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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)}">

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<annotation encoding="application/x-tex">{\displaystyle 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)}</annotation>

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</math><img src="./e8310cc3d258c38dbec0c03330b6c8896f55ae16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:63.518ex; height:7.509ex;" alt="{\displaystyle 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)}" loading="lazy">.

The convex closure of any set function is convex over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle [0,1]^{n}}</annotation>

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</math><img src="./40160923273b7109968df994dca832b91d957bf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.871ex; height:2.843ex;" alt="{\displaystyle [0,1]^{n}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Concave_closure">Concave closure</h3></div>

Consider any vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}}</annotation>

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</math><img src="./37773dae83e9f47788a37b3c174636c06b89e590.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.362ex; height:2.843ex;" alt="{\displaystyle \mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}}" loading="lazy"> such that each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq x_{i}\leq 1}">

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<annotation encoding="application/x-tex">{\displaystyle 0\leq x_{i}\leq 1}</annotation>

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</math><img src="./f96edd7ea2da465fc4fffca76a3a6c4532bc1642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.651ex; height:2.509ex;" alt="{\displaystyle 0\leq x_{i}\leq 1}" loading="lazy">. Then the concave closure is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{+}(\mathbf {x} )=\max \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)}">

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<annotation encoding="application/x-tex">{\displaystyle f^{+}(\mathbf {x} )=\max \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)}</annotation>

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</math><img src="./4d7ab3a94b8cd39c3477b3d4d23dc98cc51da1f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:63.969ex; height:7.509ex;" alt="{\displaystyle f^{+}(\mathbf {x} )=\max \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)}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Relations_between_continuous_extensions">Relations between continuous extensions</h3></div>

For the extensions discussed above, it can be shown that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{+}(\mathbf {x} )\geq F(\mathbf {x} )\geq f^{-}(\mathbf {x} )=f^{L}(\mathbf {x} )}">

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<annotation encoding="application/x-tex">{\displaystyle f^{+}(\mathbf {x} )\geq F(\mathbf {x} )\geq f^{-}(\mathbf {x} )=f^{L}(\mathbf {x} )}</annotation>

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</math><img src="./5bd88dfe80ba1d27ce50f0181e3ba2cf150a72a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.252ex; height:3.176ex;" alt="{\displaystyle f^{+}(\mathbf {x} )\geq F(\mathbf {x} )\geq f^{-}(\mathbf {x} )=f^{L}(\mathbf {x} )}" loading="lazy"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is submodular.<a href="#cite_note-JV2-12">[12]</a>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<ol><li>The class of submodular functions is <a href="Closure_(mathematics)" title="Closure (mathematics)">closed</a> under non-negative <a href="Linear_combination" title="Linear combination">linear combinations</a>. Consider any submodular function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1},f_{2},\ldots ,f_{k}}">

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</math><img src="./d638f2e6d46ec29cf80847e723577c3b38fa33e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.827ex; height:2.509ex;" alt="{\displaystyle f_{1},f_{2},\ldots ,f_{k}}" loading="lazy"> and non-negative numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{k}}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{k}}</annotation>

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</math><img src="./47898311753e759db788f62e5a4c3d0a7ce88b86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.872ex; height:2.009ex;" alt="{\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{k}}" loading="lazy">. Then the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(S)=\sum _{i=1}^{k}\alpha _{i}f_{i}(S)}">

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<annotation encoding="application/x-tex">{\displaystyle g(S)=\sum _{i=1}^{k}\alpha _{i}f_{i}(S)}</annotation>

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</math><img src="./c08c814d226516d1ab788eec467c3104a9d7412e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.8ex; height:7.343ex;" alt="{\displaystyle g(S)=\sum _{i=1}^{k}\alpha _{i}f_{i}(S)}" loading="lazy"> is submodular.</li>

<li>For any submodular function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, the function defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(S)=f(\Omega \setminus S)}">

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<annotation encoding="application/x-tex">{\displaystyle g(S)=f(\Omega \setminus S)}</annotation>

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</math><img src="./f41f474d7d1e2d1a77862ca08f4b4b617c650a3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.983ex; height:2.843ex;" alt="{\displaystyle g(S)=f(\Omega \setminus S)}" loading="lazy"> is submodular.</li>

<li>The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(S)=\min(f(S),c)}">

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<annotation encoding="application/x-tex">{\displaystyle g(S)=\min(f(S),c)}</annotation>

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</math><img src="./e371ad019492eaa3cb69ee98b14a7e2c1ff9c5a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.835ex; height:2.843ex;" alt="{\displaystyle g(S)=\min(f(S),c)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> is a real number, is submodular whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is monotone submodular. More generally, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(S)=h(f(S))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(S)=h(f(S))}</annotation>

</semantics>

</math><img src="./ce67473dd5e9c6b26ea8efef8aa24ea1cbfc8f1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.258ex; height:2.843ex;" alt="{\displaystyle g(S)=h(f(S))}" loading="lazy"> is submodular, for any non decreasing concave function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy">.</li>

<li>Consider a random process where a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> is chosen with each element in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega }</annotation>

</semantics>

</math><img src="./24b0d5ca6f381068d756f6337c08e0af9d1eeb6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Omega }" loading="lazy"> being included in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> independently with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">. Then the following inequality is true <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {E} [f(T)]\geq pf(\Omega )+(1-p)f(\varnothing )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {E} [f(T)]\geq pf(\Omega )+(1-p)f(\varnothing )}</annotation>

</semantics>

</math><img src="./bb03727a9539e45c2544a7d0df16f182042fa92f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.32ex; height:2.843ex;" alt="{\displaystyle \mathbb {E} [f(T)]\geq pf(\Omega )+(1-p)f(\varnothing )}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation>

</semantics>

</math><img src="./00595c5e33692e724937fdcc8870496acce1ac74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }" loading="lazy"> is the empty set. More generally consider the following random process where a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is constructed as follows. For each of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i\leq l,A_{i}\subseteq \Omega }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\leq i\leq l,A_{i}\subseteq \Omega }</annotation>

</semantics>

</math><img src="./7143ac90c00d0c25b3591f0f264d065a8412ea48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.208ex; height:2.509ex;" alt="{\displaystyle 1\leq i\leq l,A_{i}\subseteq \Omega }" loading="lazy"> construct <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{i}}</annotation>

</semantics>

</math><img src="./de6e810a93f67802ecb603ee0e3324005c6e583e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}" loading="lazy"> by including each element in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation>

</semantics>

</math><img src="./1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}" loading="lazy"> independently into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{i}}</annotation>

</semantics>

</math><img src="./de6e810a93f67802ecb603ee0e3324005c6e583e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}" loading="lazy"> with probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{i}}</annotation>

</semantics>

</math><img src="./5bab39399bf5424f25d957cdc57c84a0622626d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}" loading="lazy">. Furthermore let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\cup _{i=1}^{l}S_{i}}">

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S=\cup _{i=1}^{l}S_{i}}</annotation>

</semantics>

</math><img src="./8dd464ad4ecb7fb62e82d7814951372c3da4b192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.273ex; height:3.176ex;" alt="{\displaystyle S=\cup _{i=1}^{l}S_{i}}" loading="lazy">. Then the following inequality is true <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \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})}">

</mrow>

</mrow>

</munder>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \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})}</annotation>

</semantics>

</math><img src="./3b3a48254f85f96d5b3e2d8c260ec0522c911838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:44.695ex; height:6.009ex;" alt="{\displaystyle \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})}" loading="lazy">.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Optimization_problems">Optimization problems</h2></div>

Submodular functions have properties which are very similar to <a href="Convex_function" title="Convex function">convex</a> and <a href="Concave_function" title="Concave function">concave functions</a>. For this reason, an <a href="Optimization_problem" title="Optimization problem">optimization problem</a> 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.

<div class="mw-heading mw-heading3"><h3 id="Submodular_set_function_minimization">Submodular set function minimization</h3></div>

The hardness of minimizing a submodular set function depends on constraints imposed on the problem.

<ol><li>The unconstrained problem of minimizing a submodular function is computable in <a href="Polynomial_time" class="mw-redirect" title="Polynomial time">polynomial time</a>,<a href="#cite_note-GLS-13">[13]</a><a href="#cite_note-Cunningham-14">[14]</a> and even in <a href="Strongly_polynomial" class="mw-redirect" title="Strongly polynomial">strongly-polynomial</a> time.<a href="#cite_note-IFF-15">[15]</a><a href="#cite_note-Schrijver-16">[16]</a> Computing the <a href="Minimum_cut" title="Minimum cut">minimum cut</a> in a graph is a special case of this minimization problem.</li>

<li>The problem of minimizing a submodular function with a cardinality lower bound is <a href="NP-hard" class="mw-redirect" title="NP-hard">NP-hard</a>, with polynomial factor lower bounds on the approximation factor.<a href="#cite_note-SF-17">[17]</a><a href="#cite_note-IJB-18">[18]</a></li></ol>

<div class="mw-heading mw-heading3"><h3 id="Submodular_set_function_maximization">Submodular set function maximization</h3></div>

Unlike the case of minimization, maximizing a generic submodular function is <a href="NP-hard" class="mw-redirect" title="NP-hard">NP-hard</a> even in the unconstrained setting. Thus, most of the works in this field are concerned with polynomial-time approximation algorithms, including <a href="Greedy_algorithm" title="Greedy algorithm">greedy algorithms</a> or <a href="Local_search_(optimization)" title="Local search (optimization)">local search algorithms</a>.

<ol><li>The problem of maximizing a non-negative submodular function admits a 1/2 approximation algorithm.<a href="#cite_note-FMV-19">[19]</a><a href="#cite_note-BFNS-20">[20]</a> Computing the <a href="Maximum_cut" title="Maximum cut">maximum cut</a> of a graph is a special case of this problem.</li>

<li>The problem of maximizing a monotone submodular function subject to a cardinality constraint admits a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-1/e}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-1/e}</annotation>

</semantics>

</math><img src="./7dd1fc96e030c95a830e901111e6e221943d95eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.411ex; height:2.843ex;" alt="{\displaystyle 1-1/e}" loading="lazy"> approximation algorithm.<a href="#cite_note-NVF-21">[21]</a><a href="#cite_note-22">[22]</a> The <a href="Maximum_coverage_problem" title="Maximum coverage problem">maximum coverage problem</a> is a special case of this problem.</li>

<li>The problem of maximizing a monotone submodular function subject to a <a href="Matroid" title="Matroid">matroid</a> constraint (which subsumes the case above) also admits a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-1/e}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-1/e}</annotation>

</semantics>

</math><img src="./7dd1fc96e030c95a830e901111e6e221943d95eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.411ex; height:2.843ex;" alt="{\displaystyle 1-1/e}" loading="lazy"> approximation algorithm.<a href="#cite_note-CCPV-23">[23]</a><a href="#cite_note-FNS-24">[24]</a><a href="#cite_note-FW-25">[25]</a></li></ol>

Many of these algorithms can be unified within a semi-differential based framework of algorithms.<a href="#cite_note-IJB-18">[18]</a>

<div class="mw-heading mw-heading3"><h3 id="Related_optimization_problems">Related optimization problems</h3></div>

Apart from submodular minimization and maximization, there are several other natural optimization problems related to submodular functions.

<ol><li>Minimizing the difference between two submodular functions<a href="#cite_note-NB-26">[26]</a> is not only NP hard, but also inapproximable.<a href="#cite_note-IBUAI-27">[27]</a></li>

<li>Minimization/maximization of a submodular function subject to a submodular level set constraint (also known as submodular optimization subject to submodular cover or submodular knapsack constraint) admits bounded approximation guarantees.<a href="#cite_note-IB-28">[28]</a></li>

<li>Partitioning data based on a submodular function to maximize the average welfare is known as the submodular welfare problem, which also admits bounded approximation guarantees (see <a href="Welfare_maximization" title="Welfare maximization">welfare maximization</a>).</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

Submodular functions naturally occur in several real world applications, in <a href="Economics" title="Economics">economics</a>, <a href="Game_theory" title="Game theory">game theory</a>, <a href="Machine_learning" title="Machine learning">machine learning</a> and <a href="Computer_vision" title="Computer vision">computer vision</a>.<a href="#cite_note-KG-4">[4]</a><a href="#cite_note-JB-29">[29]</a> 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.

<ul><li><a href="Supermodular_function" title="Supermodular function">Supermodular function</a></li>

<li><a href="Matroid" title="Matroid">Matroid</a>, <a href="Polymatroid" title="Polymatroid">Polymatroid</a></li>

<li><a href="Utility_functions_on_indivisible_goods" title="Utility functions on indivisible goods">Utility functions on indivisible goods</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2></div>

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</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;">

<li id="cite_note-LB-1"><a href="#cite_ref-LB_1-0">^</a> H. Lin and J. Bilmes, A Class of Submodular Functions for Document Summarization, ACL-2011.

</li>

<li id="cite_note-TIWB-2"><a href="#cite_ref-TIWB_2-0">^</a> S. Tschiatschek, R. Iyer, H. Wei and J. Bilmes, Learning Mixtures of Submodular Functions for Image Collection Summarization, NIPS-2014.

</li>

<li id="cite_note-KG1-3"><a href="#cite_ref-KG1_3-0">^</a> A. Krause and C. Guestrin, Near-optimal nonmyopic value of information in graphical models, UAI-2005.

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<li id="cite_note-KG-4">^ <a href="#cite_ref-KG_4-0">a</a> <a href="#cite_ref-KG_4-1">b</a> A. Krause and C. Guestrin, Beyond Convexity: Submodularity in Machine Learning, Tutorial at ICML-2008

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> (Schrijver <a href="#CITEREFSchrijver2003">2003</a>, §44, p. 766)

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/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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<li id="cite_note-F22-8"><a href="#cite_ref-F22_8-0">^</a> Fujishige (2005) p.22

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFSchrijver2003" class="citation cs2"><a href="Alexander_Schrijver" title="Alexander Schrijver">Schrijver, Alexander</a> (2003), Combinatorial Optimization, <a href="Springer_Publishing" title="Springer Publishing">Springer</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-44389-4</bdi></cite></li>

<li><cite id="CITEREFLee2004" class="citation cs2"><a href="Jon_Lee_(mathematician)" title="Jon Lee (mathematician)">Lee, Jon</a> (2004), A First Course in Combinatorial Optimization, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-01012-8</bdi></cite></li>

<li><cite id="CITEREFFujishige2005" class="citation cs2">Fujishige, Satoru (2005), Submodular Functions and Optimization, <a href="Elsevier" title="Elsevier">Elsevier</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-444-52086-4</bdi></cite></li>

<li><cite id="CITEREFNarayanan1997" class="citation cs2">Narayanan, H. (1997), Submodular Functions and Electrical Networks, Elsevier, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-444-82523-1</bdi></cite></li>

<li><cite id="CITEREFOxley1992" class="citation cs2">Oxley, James G. (1992), Matroid theory, Oxford Science Publications, Oxford: <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-19-853563-5</bdi>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0784.05002">0784.05002</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external free" href="http://www.cs.berkeley.edu/~stefje/references.html">http://www.cs.berkeley.edu/~stefje/references.html</a> has a longer bibliography</li>

<li><a rel="nofollow" class="external free" href="http://submodularity.org/">http://submodularity.org/</a> includes further material on the subject</li></ul></div><div><div class="zim-footer">

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