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Concept# Submodular set function

Summary

In mathematics, a submodular set function (also known as a submodular function) is a set function whose value, informally, has the property that the difference in the incremental value of the function that a single element makes when added to an input set decreases as the size of the input set increases. Submodular functions have a natural diminishing returns property which makes them suitable for many applications, including approximation algorithms, game theory (as functions modeling user preferences) and electrical networks. Recently, submodular functions have also found immense utility in several real world problems in machine learning and artificial intelligence, including automatic summarization, multi-document summarization, feature selection, active learning, sensor placement, image collection summarization and many other domains.
If is a finite set, a submodular function is a set function , where denotes the power set of , which satisfies one of the following equivalent conditions.
For every with and every we have that .
For every we have that .
For every and such that we have that .
A nonnegative submodular function is also a subadditive function, but a subadditive function need not be submodular.
If is not assumed finite, then the above conditions are not equivalent. In particular a function
defined by if is finite and if is infinite
satisfies the first condition above, but the second condition fails when and are infinite sets with finite intersection.
A set function is monotone if for every we have that . Examples of monotone submodular functions include:
Linear (Modular) functions Any function of the form is called a linear function. Additionally if then f is monotone.
Budget-additive functions Any function of the form for each and is called budget additive.
Coverage functions Let be a collection of subsets of some ground set . The function for is called a coverage function. This can be generalized by adding non-negative weights to the elements.
Entropy Let be a set of random variables.

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