Submodular set function

Definition

If Ω is a finite set, a submodular function is a set function f : 2 Ω → R , where 2 Ω denotes the power set of Ω , which satisfies one of the following equivalent definitions.

1. For every X , Y ⊆ Ω with X ⊆ Y and every x ∈ Ω ∖ Y we have that f ( X ∪ { x } ) − f ( X ) ≥ f ( Y ∪ { x } ) − f ( Y ) .
2. For every S , T ⊆ Ω we have that f ( S ) + f ( T ) ≥ f ( S ∪ T ) + f ( S ∩ T ) .
3. For every X ⊆ Ω and x 1 , x 2 ∈ Ω ∖ X we have that f ( X ∪ { x 1 } ) + f ( X ∪ { x 2 } ) ≥ f ( X ∪ { x 1 , x 2 } ) + f ( X ) .

A nonnegative submodular function is also a subadditive function, but a subadditive function need not be submodular. If Ω 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.

Monotone

A submodular function f is monotone if for every T ⊆ S we have that f ( T ) ≤ f ( S ) . Examples of monotone submodular functions include:

Linear functions 

Any function of the form f ( S ) = ∑ i ∈ S w i is called a linear function. Additionally if ∀ i , w i ≥ 0 then f is monotone.

Budget-additive functions 

Any function of the form f ( S ) = min ( B , ∑ i ∈ S w i ) for each w i ≥ 0 and B ≥ 0 is called budget additive.

Coverage functions 

Let Ω = { E 1 , E 2 , … , E n } be a collection of subsets of some ground set Ω ′ . The function f ( S ) = | ∪ E i ∈ S E i | for S ⊆ Ω is called a coverage function. This can be generalized by adding non-negative weights to the elements.

Entropy 

Let Ω = { X 1 , X 2 , … , X n } be a set of random variables. Then for any S ⊆ Ω we have that H ( S ) is a submodular function, where H ( S ) is the entropy of the set of random variables S .

Matroid rank functions 

Let Ω = { e 1 , e 2 , … , 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 which is not monotone is called non-monotone.

Symmetric

A non-monotone submodular function f is called symmetric if for every S ⊆ Ω we have that f ( S ) = f ( Ω − S ) . Examples of symmetric non-monotone submodular functions include:

Graph cuts 

Let Ω = { v 1 , v 2 , … , v n } be the vertices of a graph. For any set of vertices S ⊆ Ω let f ( S ) denote the number of edges e = ( u , v ) such that u ∈ S and v ∈ Ω − S . This can be generalized by adding non-negative weights to the edges.

Mutual information 

Let Ω = { X 1 , X 2 , … , X n } be a set of random variables. Then for any S ⊆ Ω we have that f ( S ) = I ( S ; Ω − S ) is a submodular function, where I ( S ; Ω − S ) is the mutual information.

Asymmetric

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

Directed cuts 

Let Ω = { v 1 , v 2 , … , v n } be the vertices of a directed graph. For any set of vertices S ⊆ Ω let f ( S ) denote the number of edges e = ( u , v ) such that u ∈ S and v ∈ Ω − S . This can be generalized by adding non-negative weights to the directed edges.

Lovász extension

This extension is named after mathematician László Lovász. Consider any vector x = { x 1 , x 2 , … , x n } such that each 0 ≤ x i ≤ 1 . Then the Lovász extension is defined as f L ( x ) = E ( f ( { i | x i ≥ λ } ) ) where the expectation is over λ chosen from the uniform distribution on the interval [ 0 , 1 ] . The Lovász extension is a convex function.

Multilinear extension

Consider any vector x = { x 1 , x 2 , … , x n } such that each 0 ≤ x i ≤ 1 . Then the multilinear extension is defined as F ( x ) = ∑ S ⊆ Ω f ( S ) ∏ i ∈ S x i ∏ i ∉ S ( 1 − x i ) .

Convex closure

Consider any vector x = { x 1 , x 2 , … , x n } such that each 0 ≤ x i ≤ 1 . Then the convex closure is defined as f − ( x ) = min ( ∑ S α S f ( S ) : ∑ S α S 1 S = x , ∑ S α S = 1 , α S ≥ 0 ) . It can be shown that f L ( x ) = f − ( x ) .

Concave closure

Consider any vector x = { x 1 , x 2 , … , x n } such that each 0 ≤ x i ≤ 1 . Then the concave closure is defined as f + ( x ) = max ( ∑ S α S f ( S ) : ∑ S α S 1 S = x , ∑ S α S = 1 , α S ≥ 0 ) .

Properties

1. The class of submodular functions is closed under non-negative linear combinations. Consider any submodular function f 1 , f 2 , … , f k and non-negative numbers α 1 , α 2 , … , α k . Then the function g defined by g ( S ) = ∑ i = 1 k α i f i ( S ) is submodular. Furthermore, for any submodular function f , the function defined by g ( S ) = f ( Ω ∖ S ) is submodular. The function g ( S ) = min ( f ( S ) , c ) , where c is a real number, is submodular whenever f is monotonic.
2. If f : 2 Ω → R + is a submodular function then g : 2 Ω → R + defined as g ( S ) = ϕ ( f ( S ) ) where ϕ is a concave function, is also a submodular function.
3. Consider a random process where a set T is chosen with each element in Ω being included in T independently with probability p . Then the following inequality is true E [ f ( T ) ] ≥ p f ( Ω ) + ( 1 − p ) f ( ∅ ) where ∅ is the empty set. More generally consider the following random process where a set S is constructed as follows. For each of 1 ≤ i ≤ l , A i ⊆ Ω construct S i by including each element in A i independently into S i with probability p i . Furthermore let S = ∪ i = 1 l S i . Then the following inequality is true E [ f ( S ) ] ≥ ∑ R ⊆ [ l ] Π i ∈ R p i Π i ∉ R ( 1 − p i ) f ( ∪ i ∈ 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 Minimization

The simplest minimization problem is to find a set S ⊆ Ω 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. However, even simple constraints like cardinality lower bound constraints make this problem NP hard, with polynomial lower bound approximation factors.

Submodular Maximization

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 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 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