Submodular set 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.

Definition

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

1. For every $X,Y\subseteq \Omega$ with $X\subseteq Y$ and every $x\in \Omega \setminus Y$ we have that $f(X\cup \{x\})-f(X)\geq f(Y\cup \{x\})-f(Y)$.
2. For every $S,T\subseteq \Omega$ we have that $f(S)+f(T)\geq f(S\cup T)+f(S\cap T)$.
3. For every $X\subseteq \Omega$ and $x_{1},x_{2}\in \Omega \backslash X$ such that $x_{1}\neq x_{2}$ we have that $f(X\cup \{x_{1}\})+f(X\cup \{x_{2}\})\geq f(X\cup \{x_{1},x_{2}\})+f(X)$.

A nonnegative submodular function is also a subadditive function, but a subadditive function need not be submodular. If $\Omega$ is not assumed finite, then the above conditions are not equivalent. In particular a function $f$ defined by $f(S)=1$ if $S$ is finite and $f(S)=0$ if $S$ is infinite satisfies the first condition above, but the second condition fails when $S$ and $T$ are infinite sets with finite intersection.

Types and examples of submodular functions

Monotone

A submodular function $f$ is monotone if for every $T\subseteq S$ we have that $f(T)\leq f(S)$. Examples of monotone submodular functions include:

Linear (Modular) functions

Any function of the form $f(S)=\sum _{i\in S}w_{i}$ is called a linear function. Additionally if $\forall i,w_{i}\geq 0$ then f is monotone.

Budget-additive functions

Any function of the form $f(S)=\min \left\{B,~\sum _{i\in S}w_{i}\right\}$ for each $w_{i}\geq 0$ and $B\geq 0$ is called budget additive.

Coverage functions

Let $\Omega =\{E_{1},E_{2},\ldots ,E_{n}\}$ be a collection of subsets of some ground set $\Omega '$. The function $f(S)=\left|\bigcup _{E_{i}\in S}E_{i}\right|$ for $S\subseteq \Omega$ is called a coverage function. This can be generalized by adding non-negative weights to the elements.

Entropy

Let $\Omega =\{X_{1},X_{2},\ldots ,X_{n}\}$ be a set of random variables. Then for any $S\subseteq \Omega$ we have that $H(S)$ is a submodular function, where $H(S)$ is the entropy of the set of random variables $S$, a fact known as Shannon's inequality. Further inequalities for the entropy function are known to hold, see entropic vector.

Matroid rank functions

Let $\Omega =\{e_{1},e_{2},\dots ,e_{n}\}$ be the ground set on which a matroid is defined. Then the rank function of the matroid is a submodular function.

Non-monotone

A submodular function that is not monotone is called non-monotone.

Symmetric

A non-monotone submodular function $f$ is called symmetric if for every $S\subseteq \Omega$ we have that $f(S)=f(\Omega -S)$. Examples of symmetric non-monotone submodular functions include:

Graph cuts

Let $\Omega =\{v_{1},v_{2},\dots ,v_{n}\}$ be the vertices of a graph. For any set of vertices $S\subseteq \Omega$ let $f(S)$ denote the number of edges $e=(u,v)$ such that $u\in S$ and $v\in \Omega -S$. This can be generalized by adding non-negative weights to the edges.

Mutual information

Let $\Omega =\{X_{1},X_{2},\ldots ,X_{n}\}$ be a set of random variables. Then for any $S\subseteq \Omega$ we have that $f(S)=I(S;\Omega -S)$ is a submodular function, where $I(S;\Omega -S)$ is the mutual information.

Asymmetric

A non-monotone submodular function which is not symmetric is called asymmetric.

Directed cuts

Let $\Omega =\{v_{1},v_{2},\dots ,v_{n}\}$ be the vertices of a directed graph. For any set of vertices $S\subseteq \Omega$ let $f(S)$ denote the number of edges $e=(u,v)$ such that $u\in S$ and $v\in \Omega -S$. This can be generalized by adding non-negative weights to the directed edges.

Continuous extensions

Generic definition

For a submodular function $f:2^{\Omega }\rightarrow \mathbb {R}$ with $|\Omega |=n$, we can associate each $S\subseteq \Omega$ with a binary vector $x^{S}\in \{0,1\}^{n}$ such that $x_{i}^{S}=1$ when $i\in S$, and $x_{i}^{S}=0$ otherwise. In this sense, the submodular function can be seen as a function defined on $\{0,1\}^{n}$. We can then define the continuous extension of $f$ to be any continuous function $F:\mathbb {R} ^{n}\rightarrow \mathbb {R}$ such that it matches the value of the submodular function on $x\in \{0,1\}^{n}$, i.e. $F(x^{S})=f(S)$. Some commonly used continuous extensions are as follows.

Lovász extension

This extension is named after mathematician László Lovász. Consider any vector $\mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}$ such that each $0\leq x_{i}\leq 1$. Then the Lovász extension is defined as $f^{L}(\mathbf {x} )=\mathbb {E} (f(\{i|x_{i}\geq \lambda \}))$ where the expectation is over $\lambda$ chosen from the uniform distribution on the interval $[0,1]$. The Lovász extension is a convex function if and only if $f$ is a submodular function.

Multilinear extension

Consider any vector $\mathbf {x} =\{x_{1},x_{2},\ldots ,x_{n}\}$ such that each $0\leq x_{i}\leq 1$. Then the multilinear extension is defined as $F(\mathbf {x} )=\sum _{S\subseteq \Omega }f(S)\prod _{i\in S}x_{i}\prod _{i\notin S}(1-x_{i})$.

Convex closure

Consider any vector $\mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}$ such that each $0\leq x_{i}\leq 1$. Then the convex closure is defined as $f^{-}(\mathbf {x} )=\min \left(\sum _{S}\alpha _{S}f(S):\sum _{S}\alpha _{S}1_{S}=\mathbf {x} ,\sum _{S}\alpha _{S}=1,\alpha _{S}\geq 0\right)$. The convex closure of any set function is convex over $[0,1]^{n}$. It can be shown that $f^{L}(\mathbf {x} )=f^{-}(\mathbf {x} )$ for submodular functions.

Concave closure

Consider any vector $\mathbf {x} =\{x_{1},x_{2},\dots ,x_{n}\}$ such that each $0\leq x_{i}\leq 1$. Then the concave closure is defined as $f^{+}(\mathbf {x} )=\max \left(\sum _{S}\alpha _{S}f(S):\sum _{S}\alpha _{S}1_{S}=\mathbf {x} ,\sum _{S}\alpha _{S}=1,\alpha _{S}\geq 0\right)$.

Properties

1. The class of submodular functions is closed under non-negative linear combinations. Consider any submodular function $f_{1},f_{2},\ldots ,f_{k}$ and non-negative numbers $\alpha _{1},\alpha _{2},\ldots ,\alpha _{k}$. Then the function $g$ defined by $g(S)=\sum _{i=1}^{k}\alpha _{i}f_{i}(S)$ is submodular.
2. For any submodular function $f$, the function defined by $g(S)=f(\Omega \setminus S)$ is submodular.
3. The function $g(S)=\min(f(S),c)$, where $c$ is a real number, is submodular whenever $f$ is monotone submodular. More generally, $g(S)=h(f(S))$ is submodular, for any non decreasing concave function $h$.
4. Consider a random process where a set $T$ is chosen with each element in $\Omega$ being included in $T$ independently with probability $p$. Then the following inequality is true $\mathbb {E} [f(T)]\geq pf(\Omega )+(1-p)f(\varnothing )$ where $\varnothing$ is the empty set. More generally consider the following random process where a set $S$ is constructed as follows. For each of $1\leq i\leq l,A_{i}\subseteq \Omega$ construct $S_{i}$ by including each element in $A_{i}$ independently into $S_{i}$ with probability $p_{i}$. Furthermore let $S=\cup _{i=1}^{l}S_{i}$. Then the following inequality is true $\mathbb {E} [f(S)]\geq \sum _{R\subseteq [l]}\Pi _{i\in R}p_{i}\Pi _{i\notin R}(1-p_{i})f(\cup _{i\in R}A_{i})$.

Optimization problems

Submodular functions have properties which are very similar to convex and concave functions. For this reason, an optimization problem which concerns optimizing a convex or concave function can also be described as the problem of maximizing or minimizing a submodular function subject to some constraints.

Submodular set function minimization

The simplest minimization problem is to find a set $S\subseteq \Omega$ which minimizes a submodular function; this is the unconstrained problem. This problem is computable in (strongly) polynomial time. Computing the minimum cut in a graph is a special case of this general minimization problem. However, adding even a simple constraint such as a cardinality lower bound makes the minimization problem NP hard, with polynomial factor lower bounds on the approximation factor.

Submodular set function maximization

Unlike the case of minimization, maximizing a submodular functions is NP-hard even in the unconstrained setting. Theory and enumeration algorithms for finding local and global maxima (minima) of submodular (supermodular) functions can be found in B. Goldengorin. European Journal of Operational Research 198(1):102-112, DOI: 10.1016/j.ejor.2008.08.022. For instance max cut is a special case even when the function is required only to be non-negative. The unconstrained problem can be shown to be inapproximable if it is allowed to be negative. There has been extensive work on constrained submodular function maximization when the functions are non-negative. Typically, the approximation algorithms for these problems are based on either greedy algorithms or local search algorithms. The problem of maximizing a non-negative symmetric submodular function admits a 1/2 approximation algorithm. Computing the maximum cut of a graph is a special case of this problem. The more general problem of maximizing a non-negative submodular function also admits a 1/2 approximation algorithm. The problem of maximizing a monotone submodular function subject to a cardinality constraint admits a $1-1/e$ approximation algorithm. The maximum coverage problem is a special case of this problem. The more general problem of maximizing a monotone submodular function subject to a matroid constraint also admits a $1-1/e$ approximation algorithm. Many of these algorithms can be unified within a semi-differential based framework of algorithms.

Related optimization problems

Apart from submodular minimization and maximization, another natural problem is Difference of Submodular Optimization. Unfortunately, this problem is not only NP hard, but also inapproximable. A related optimization problem is minimize or maximize a submodular function, subject to a submodular level set constraint (also called submodular optimization subject to submodular cover or submodular knapsack constraint). This problem admits bounded approximation guarantees. Another optimization problem involves partitioning data based on a submodular function, so as to maximize the average welfare. This problem is called the submodular welfare problem.

Applications

Submodular functions naturally occur in several real world applications, in economics, game theory, machine learning and computer vision. 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. For more information on applications of submodularity, particularly in machine learning, see

See also

• Supermodular function
• Matroid, Polymatroid
• Utility functions on indivisible goods

References

• Schrijver, Alexander (2003), Combinatorial Optimization, Springer, ISBN 3-540-44389-4
• Lee, Jon (2004), A First Course in Combinatorial Optimization, Cambridge University Press, ISBN 0-521-01012-8
• Fujishige, Satoru (2005), Submodular Functions and Optimization, Elsevier, ISBN 0-444-52086-4
• Narayanan, H. (1997), Submodular Functions and Electrical Networks, ISBN 0-444-82523-1
• Oxley, James G. (1992), Matroid theory, Oxford Science Publications, Oxford: Oxford University Press, ISBN 0-19-853563-5, Zbl 0784.05002

External links

• http://www.cs.berkeley.edu/~stefje/references.html has a longer bibliography