Subadditive set function

In mathematics, a subadditive set function is a set function whose value, informally, has the property that the value of function on the union of two sets is at most the sum of values of the function on each of the sets. This is thematically related to the subadditivity property of real-valued functions.

Definition

If \Omega is a set, a subadditive function is a set function f:2^{\Omega}\rightarrow \mathbb{R}, where 2^\Omega denotes the power set of \Omega, which satisfies the following inequality.[1][2]

For every S, T \subseteq \Omega we have that f(S)+f(T)\geq f(S\cup T).

Examples of subadditive functions

1. Every non-negative submodular set function is subadditive (the family of submodular functions is strictly contained in the family of subadditive functions).

2. Functions based on set cover. Let T_1,T_2,\ldots,T_m\subseteq \Omega such that \cup_{i=1}^m T_i=\Omega. Then f is defined as the minimum number of subsets required to cover a given set:            f(S)=\min t such that there exists sets T_{i_1},T_{i_2},\dots,T_{i_t} satisfying S\subseteq \cup_{j=1}^t T_{i_j}.

3. The maximum of additive set functions is subadditive (Dually, the minimum of additive functions is superadditive). Formally, let for each i\in \{1,2,\ldots,m\}, a_i:\Omega\rightarrow \mathbb{R}_+ be linear (additive) set functions. Then f(S)=\max_{i}\left(\sum_{x\in S}a_i(x)\right) is a subadditive set function.

4. Fractionally subadditive set functions. This is a generalization of submodular function and special case of subadditive function. If \Omega is a set, a fractionally subadditive function is a set function f:2^{\Omega}\rightarrow \mathbb{R}, where 2^\Omega denotes the power set of \Omega, which satisfies the following definition:[1]

Properties

  1. If T is a set chosen such that each e\in \Omega is included into T with probability p then the following inequality is satisfied \mathbb{E}[f(T)]\geq p f(\Omega).

See also

Citations

  1. ↑ 1.0 1.1 1.2 U. Feige, On Maximizing Welfare when Utility Functions are Subadditive, SIAM J. Comput 39 (2009), pp. 122–142.
  2. ↑ S. Dobzinski, N. Nisan,M. Schapira, Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders, Math. Oper. Res. 35 (2010), pp. 1–13.