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Mathreader17 (talk | contribs) add a generic introduction to continuous extensions |
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== Continuous extensions ==
For a submodular function <math>f:2^{\Omega}\rightarrow \mathbb{R}</math> with <math>|\Omega|=n</math>, we can associate each <math>S\subseteq \Omega</math> with a binary vector <math>x^{S}\in \{0, 1\}^{n}</math> such that <math>x_{i}^{S}=1</math> when <math>i\in S</math>, and <math>x_{i}^{S}=0</math> otherwise. In this sense, the submodular function can be seen as a function defined on <math>\{0, 1\}^{n}</math>. We can then define the continuous [[Restriction_(mathematics)#Extension_of_a_function|extension]] of <math>f</math> to be any continuous function <math>F:\mathbb{R}^{n}\rightarrow \mathbb{R}</math> such that it matches the value of the submodular function on <math>x\in \{0, 1\}^{n}</math>, i.e. <math>F(x^{S})=f(S)</math>. Some commonly used continuous extensions are as follows.
=== Lovász extension ===
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