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 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 and game theory (as functions modeling user preferences).
- For every with and every we have that .
- For every we have that .
- For every and we have that .
A nonnegative submodular function is also a subadditive function, but a subadditive function need not be submodular.
Types of submodular functions
A submodular function is monotone if for every we have that . Examples of monotone submodular functions include:
- Linear functions
- Any function of the form is called a linear function. Additionally if then f is monotone.
- Budget-additive functions
- Any function of the form for each and is called budget additive.
- Coverage functions
- Let be a collection of subsets of some ground set . The function for is called a coverage function. This can be generalized by adding non-negative weights to the elements.
- Let be a set of random variables. Then for any we have that is a submodular function, where is the entropy of the set of random variables
- Matroid rank functions
- Let be the ground set on which a matroid is defined. Then the rank function of the matroid is a submodular function.
A submodular function which is not monotone is called non-monotone.
A non-monotone submodular function is called symmetric if for every we have that . Examples of symmetric non-monotone submodular functions include:
- Graph cuts
- Let be the vertices of a graph. For any set of vertices let denote the number of edges such that and . This can be generalized by adding non-negative weights to the edges.
- Mutual information
- Let be a set of random variables. Then for any we have that is a submodular function, where is the mutual information.
A non-monotone submodular function which is not symmetric is called asymmetric.
- Directed cuts
- Let be the vertices of a directed graph. For any set of vertices let denote the number of edges such that and . This can be generalized by adding non-negative weights to the directed edges.
This extension is named after mathematician László Lovász. Consider any vector such that each . Then the Lovász extension is defined as where the expectation is over chosen from the uniform distribution on the interval . The Lovász extension is a convex function.
Consider any vector such that each . Then the multilinear extension is defined as .
Consider any vector such that each . Then the convex closure is defined as . It can be shown that .
Consider any vector such that each . Then the concave closure is defined as .
- The class of submodular functions is closed under non-negative linear combinations. Consider any submodular function and non-negative numbers . Then the function defined by is submodular. Furthermore, for any submodular function , the function defined by is submodular. The function , where is a real number, is submodular whenever is monotonic.
- If is a submodular function then defined as where is a concave function, is also a submodular function.
- Consider a random process where a set is chosen with each element in being included in independently with probability . Then the following inequality is true where is the empty set. More generally consider the following random process where a set is constructed as follows. For each of construct by including each element in independently into with probability . Furthermore let . Then the following inequality is true .
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.
The simplest minimization problem is to find a set which minimizes a submodular function subject to no constraints. This problem is computable in (strongly) polynomial time. Computing the minimum cut in a graph is a special case of this general minimization problem.
Unlike minimization, maximization of submodular functions is usually NP-hard. Many problems, such as max cut and the maximum coverage problem, can be cast as special cases of this general maximization problem under suitable constraints. Typically, the approximation algorithms for these problems are based on either greedy algorithms or local search algorithms. The problem of maximizing a symmetric non-monotone submodular function subject to no constraints 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 an arbitrary non-monotone submodular function subject to no constraints also admits a 1/2 approximation algorithm. The problem of maximizing a monotone submodular function subject to a cardinality constraint admits a 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 approximation algorithm.
- (Schrijver 2003, §44, p. 766)
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