Submodular function

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 ${\displaystyle \Omega }$ is a finite set, a submodular function is a set function ${\displaystyle f:2^{\Omega }\rightarrow \mathbb {R} }$, where ${\displaystyle 2^{\Omega }}$ denotes the power set of ${\displaystyle \Omega }$, which satisfies one of the following equivalent definitions.

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