Submodular set function: Difference between revisions

Revision as of 21:50, 25 September 2023

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== Continuous extensions of submodular set functions ==

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>f:2^{\Omega}\rightarrow \mathbb{R}</math> with <math>|\Omega|=n</math> can be represented as a function on <math>\{0, 1\}^{n}</math>, by associating 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. A ''continuous [[Restriction_Restriction (mathematics)#Extension_of_a_functionExtension of a function|extension]]'' of <math>f</math> is a continuous function <math>F:[0, 1]^{n}\rightarrow \mathbb{R}</math>, that matches the value of <math>f</math> on <math>x\in \{0, 1\}^{n}</math>, i.e. <math>F(x^{S})=f(S)</math>.

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

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This extension is named after mathematician [[László Lovász]].<ref name="L" /> Consider any vector <math>\mathbf{x}=\{x_1,x_2,\dots,x_n\}</math> such that each <math>0\leq x_i\leq 1</math>. Then the Lovász extension is defined as

<math>f^L(\mathbf{x})=\mathbb{E}(f(\{i|x_i\geq \lambda\}))</math>

where the expectation is over <math>\lambda</math> chosen from the [[uniform distribution (continuous)|uniform distribution]] on the interval <math>[0,1]</math>. The Lovász extension is a convex function if and only if <math>f</math> is a submodular function.

=== 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 journal |last=Vondrak |first=Jan |date=2008-05-17 |title=Optimal approximation for the submodular welfare problem in the value oracle model |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}}</ref><ref>{{Cite journal |last=Calinescu |first=Gruia |last2=Chekuri |first2=Chandra |last3=Pál |first3=Martin |last4=Vondrák |first4=Jan |date=January 2011-01 |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}}</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.

=== Convex closure ===

Consider any vector <math>\mathbf{x}=\{x_1,x_2,\dots,x_n\}</math> such that each <math>0\leq x_i\leq 1</math>. Then the convex closure is defined as <math>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)</math>.

The convex closure of any set function is convex over <math>[0,1]^n</math>.

=== Concave closure ===

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=== Submodular set function minimization===

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

# The unconstrained problem of minimizing a submodular function is computable in [[polynomial time]],<ref name="GLS" /><ref name="Cunningham" /> and even in [[Strongly polynomial|strongly-polynomial]] time.<ref name="IFF" /><ref name="Schrijver" /> Computing the [[minimum cut]] in a graph is a special case of this minimization problem.

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== Applications ==

Submodular functions naturally occur in several real world applications, in [[economics]], [[game theory]], [[machine learning]] and [[computer vision]].<ref name="KG" /><ref name="JB" /> 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.

== See also ==