Subadditive set function: Difference between revisions

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== Definition ==

Let <math>\Omega</math> be a [[set (mathematics)|set]] and <math>f \colon 2^{\Omega} \rightarrow \mathbb{R}</math> be a [[set function]], where <math>2^\Omega</math> denotes the [[Power set#Representing subsets as functions|power set]] of <math>\Omega</math>. The function ''f'' is ''subadditive'' if for each subset <math>S</math> and <math>T</math> of <math>\Omega</math>, we have <math>f(S) + f(T) \geq f(S \cup T)</math>.<ref name="UF" /><ref name="DNS" /> Note that by substitution of <math>T=S</math> into the defining equation, it follows that <math>f(S) \ge 0</math> for all {{tmath|S}}.

== Examples of subadditive functions ==

[[Image:Principle of sigma-subadditivity.svg|thumb|Everyday example of sigma sub-additivity: when sand is mixed with water, the [[bulk volume]] of the mixture is smaller than the sum of the individual volumes, as the water can lodge in the spaces between the sand grains. A similar situation with a different mechanism of action occurs when ethanol is mixed with water, see [[volumeapparent molar contractionproperty]].]]

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The function that counts the number of sets required to [[set cover|cover]] a given set is subadditive. Let <math>T_1, \dotsc, T_m \subseteq \Omega</math> such that <math>\cup_{i=1}^m T_i=\Omega</math>. Define <math>f</math> as the minimum number of subsets required to cover a given set. Formally, <math>f(S)</math> is the minimum number <math>t</math> such that there are sets <math>T_{i_1}, \dotsc, T_{i_t}</math> satisfying <math>S\subseteq \cup_{j=1}^t T_{i_j}</math>. Then <math>f</math> is subadditive.

The [[maximum]] of [[additive map|additive set function]]s is subadditive (dually, the [[minimum]] of additive functions is [[superadditive]]). Formally, for each <math>i \in \{1, \dotsc, m\}</math>, let <math>a_i \colon \Omega \to \mathbb{R}_+</math> be additive set functions. Then <math>f(S)=\max_{i}\left(\sum_{x\in S}a_i(x)\rightS)</math> is a subadditive set function.

Fractionally subadditive set functions are a generalization of submodular functions and a special case of subadditive functions. A subadditive function <math>f</math> is furthermore fractionally subadditive if it satisfies the following definition.<ref name="UF" /> For every <math>S \subseteq \Omega</math>, every <math>X_1, \dotsc, X_n \subseteq \Omega</math>, and every <math>\alpha_1, \dotsc, \alpha_n \in [0, 1]</math>, if <math>1_S \leq \sum_{i=1}^n \alpha_i 1_{X_i}</math>, then <math>f(S) \leq \sum_{i=1}^n \alpha_i f(X_i)</math>. The set of fractionally subadditive functions equals the set of functions that can be expressed as the maximum of additive functions, as in the example in the previous paragraph.<ref name="UF" />

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refs=

<ref name="UF">{{cite journal | first=Uriel | last=Feige | authorlink=Uriel Feige | title=On Maximizing Welfare when Utility Functions are Subadditive | journal=SIAM Journal on Computing | volume=39 | issue=1 | year=2009 | pages=122–142 | doi=10.1137/070680977}}</ref>

<ref name="DNS">{{cite journal | first1=Shahar | last1=Dobzinski | first2=Noam | last2=Nisan | first3=Michael | last3=Schapira | authorlink2=Noam Nisan | title=Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders | journal=[[Mathematics of Operations Research]] | volume=35 | issue=1 | year=2010 | pages=1–13 | doi=10.1145/1060590.1060681| s2cid=2685385 | citeseerx=10.1.1.79.6803 }}</ref>

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