What is submodular maximization?

What is submodular maximization?

DR-submodularity captures a subclass of non-convex functions that enables both exact minimization and approximate maximization in polynomial time. In this work we study the problem of maximizing non-monotone DR-submodular continuous functions under general down-closed convex constraints.

Is the sum of submodular functions submodular?

Sum of submodular functions is submodular.

Is concave function submodular?

Submodular functions also show signs similar to concavity. Submodular function maximization, though NP-hard, admits constant-factor approximation guarantees, and concave functions composed with modular functions are submodular.

Are linear functions submodular?

More generally, αf + βg is submodular for any α, β ≥ 0. If g is a linear function then −g is also linear and hence f −g is submodular if f is submodular and g is linear. A useful context in which this arises is the following.

How do you prove Submodularity?

Lemma 1 A function f : 2N → R is submodular if and only if for every X, Y ⊆ N, f(X) + f(Y ) ≥ f(X ∩ Y ) + f(X ∪ Y ). by the vertex set S. It is an easy exercise to prove that f is a submodular function.

Is a set a function?

A set function is a function whose domain is a collection of sets. In many instances in real analysis, a set function is a function which associates an affinely extended real number to each set in a collection of sets.

Is Submodular function convex?

In particular, we show how submodular function minimization is equivalent to solving a wide variety of convex optimization problems….Learning with Submodular Functions: A Convex Optimization Perspective.

How do you prove a function is submodular?

A function f : 2N → R is said to be submodular, if it satisfies following property of diminishing marginal returns: for every A ⊆ B ⊆ N and j ∈ B, f(A ∪ {j}) − f(A) ≥ f(B ∪ {j}) − f(B). One way to understand submodularity is to think of f as a utility functions.

What are combinatorial optimization problems?

Formal definition Formally, a combinatorial optimization problem is a quadruple , where. is a set of instances; given an instance , is the finite set of feasible solutions; given an instance and a feasible solution of , denotes the measure of. , which is usually a positive real.

Is Submodularity testable?

Clearly, if the input function is submodular, the tester answers YES.

How do you know if it’s not a function?

Use the vertical line test to determine whether or not a graph represents a function. If a vertical line is moved across the graph and, at any time, touches the graph at only one point, then the graph is a function. If the vertical line touches the graph at more than one point, then the graph is not a function.

How useful is combinatorial optimization?

In the modern era, combinatorial optimization is useful for the study of algorithms, with special relevance to artificial intelligence, machine learning, and operations research.

What is combinatorial problem with example?

As an example of a combinatorial decision problem, consider the Graph Colouring Problem: given a graph G and a number of colours, find an assignment of colours to the vertices of G such that two vertices that are connected by an edge are never assigned the same colour.

How do you prove a function is Submodular?

How do you evaluate a function?

Evaluating a function means finding the value of f(x) =… or y =… that corresponds to a given value of x. To do this, simply replace all the x variables with whatever x has been assigned. For example, if we are asked to evaluate f(4), then x has been assigned the value of 4.

Why is combinatorial optimization hard?

The difficulty arises from the fact that unlike linear programming, the feasible region of the combinatorial problem is not a convex set. Thus, we must, instead, search a lattice of feasible points, or in the case of the mixed integer case, a set of disjoint half-lines or line segments to find an optimal solution.