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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).

Definition

If $Ω$ is a set, a submodular function is a set function $f : 2^{Ω} \to ℝ$ , where $2^{Ω}$ denotes the power set of $Ω$ , which satisfies one of the following equivalent definitions.^[1]

For every $X, Y \subseteq Ω$ with $X \subseteq Y$ and every $x \in Ω ∖ Y$ we have that $f (X \cup {x}) - f (X) \geq f (Y \cup {x}) - f (Y)$ .
For every $S, T \subseteq Ω$ we have that $f (S) + f (T) \geq f (S \cup T) + f (S \cap T)$ .
For every $X \subseteq Ω$ and $x_{1}, x_{2} \in Ω ∖ X$ 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.

Types 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 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 $f (S) = \min (B, \sum_{i \in S} w_{i})$ for each $w_{i} \geq 0$ and $B \geq 0$ is called budget additive.
Coverage functions: Let $Ω = {E_{1}, E_{2}, \dots, E_{n}}$ be a collection of subsets of some ground set $Ω^{'}$ . The function $f (S) = | \cup_{E_{i} \in S} E_{i} |$ for $S \subseteq Ω$ is called a coverage function. This can be generalized by adding non-negative weights to the elements.
Entropy: Let $Ω = {X_{1}, X_{2}, \dots, X_{n}}$ be a set of random variables. Then for any $S \subseteq Ω$ 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 $Ω = {e_{1}, e_{2}, \dots, e_{n}}$ 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 which is not monotone is called non-monotone.

Symmetric

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

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

Continuous extensions

Lovász extension

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

Properties

The class of submodular functions is closed under non-negative linear combinations. Consider any submodular function $f_{1}, f_{2}, \dots, f_{k}$ and non-negative numbers $α_{1}, α_{2}, \dots, α_{k}$ . Then the function $g$ defined by $g (S) = \sum_{i = 1}^{k} α_{i} f_{i} (S)$ 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.
If $f : 2^{Ω} \to ℝ_{+}$ is a submodular function then $g : 2^{Ω} \to ℝ_{+}$ defined as $g (S) = ϕ (f (S))$ where $ϕ$ is a concave function, is also a submodular function.
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 $𝔼 [f (T)] \geq p f (Ω) + (1 - p) f (\emptyset)$ where $\emptyset$ is the empty set. More generally consider the following random process where a set $S$ is constructed as follows. For each of $1 \leq i \leq l, A_{i} \subseteq Ω$ 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 $𝔼 [f (S)] \geq \sum_{R \subseteq [l]} Π_{i \in R} p_{i} Π_{i \notin R} (1 - p_{i}) f (\cup_{i \in R} A_{i})$ .Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

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.

The simplest minimization problem is to find a set $S \subseteq Ω$ which minimizes a submodular function subject to no constraints. This problem is computable in (strongly)^[2]^[3] polynomial time.^[4]^[5] 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.^[6] 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.^[7] The problem of maximizing a monotone submodular function subject to a cardinality constraint admits a $1 - 1 / e$ approximation algorithm.^[8] 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.^[9]^[10]

Citations

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References

Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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-Hi there! :) My name is Adrianna, I'm a student studying Comparative Politics from Lelystad, Netherlands.
+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).
+== Definition ==
+If <math>\Omega</math> is a [[set (mathematics)|set]], a submodular function is a set function <math>f:2^{\Omega}\rightarrow \mathbb{R}</math>, where <math>2^\Omega</math> denotes the [[Power set#Representing subsets as functions|power set]] of <math>\Omega</math>, which satisfies one of the following equivalent definitions.<ref>{{Harvard citations|last = Schrijver|year = 2003|loc = §44, p. 766|nb = }}</ref>
+# For every <math>X, Y \subseteq \Omega</math> with <math> X \subseteq Y</math> and every <math>x \in \Omega \backslash Y</math> we have that <math>f(X\cup \{x\})-f(X)\geq f(Y\cup \{x\})-f(Y)</math>.
+# For every <math>S, T \subseteq \Omega</math> we have that <math>f(S)+f(T)\geq f(S\cup T)+f(S\cap T)</math>.
+# For every <math>X\subseteq \Omega</math> and <math>x_1,x_2\in \Omega\backslash X</math> we have that <math>f(X\cup \{x_1\})+f(X\cup \{x_2\})\geq f(X\cup \{x_1,x_2\})+f(X)</math>.
+A nonnegative submodular function is also a [[Subadditive set function|subadditive]] function, but a subadditive function need not be submodular.
+== Types of submodular functions ==
+=== Monotone ===
+A submodular function <math>f</math> is ''monotone'' if for every <math>T\subseteq S</math> we have that <math>f(T)\leq f(S)</math>. Examples of monotone submodular functions include:
+; Linear functions : Any function of the form <math>f(S)=\sum_{i\in S}w_i</math> is called a linear function. Additionally if <math>\forall i,w_i\geq 0</math> then f is monotone.
+; Budget-additive functions : Any function of the form <math>f(S)=\min(B,\sum_{i\in S}w_i)</math> for each <math>w_i\geq 0</math> and <math>B\geq 0</math> is called budget additive.
+; Coverage functions : Let <math>\Omega=\{E_1,E_2,\ldots,E_n\}</math> be a collection of subsets of some ground set <math>\Omega'</math>. The function <math>f(S)=|\cup_{E_i\in S}E_i|</math> for <math>S\subseteq \Omega</math> is called a coverage function. This can be generalized by adding non-negative weights to the elements.
+; [[Entropy (information theory)|Entropy]] : Let <math>\Omega=\{X_1,X_2,\ldots,X_n\}</math> be a set of [[random variables]]. Then for any <math>S\subseteq \Omega</math> we have that <math>H(S)</math> is a submodular function, where <math>H(S)</math> is the entropy of the set of random variables <math>S</math>
+; [[Matroid]] [[matroid rank|rank functions]] : Let <math>\Omega=\{e_1,e_2,\dots,e_n\}</math> 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 <math>f</math> is called ''symmetric'' if for every <math>S\subseteq \Omega</math> we have that <math>f(S)=f(\Omega-S)</math>.
+Examples of symmetric non-monotone submodular functions include:
+; Graph cuts : Let <math>\Omega=\{v_1,v_2,\dots,v_n\}</math> be the vertices of a [[graph (mathematics)|graph]]. For any set of vertices <math>S\subseteq \Omega</math> let <math>f(S)</math> denote the number of edges <math>e=(u,v)</math> such that <math>u\in S</math> and <math>v\in \Omega-S</math>. This can be generalized by adding non-negative weights to the edges.
+; [[Mutual information]] : Let <math>\Omega=\{X_1,X_2,\ldots,X_n\}</math> be a set of [[random variable]]s. Then for any <math>S\subseteq \Omega</math> we have that <math>f(S)=I(S;\Omega-S)</math> is a submodular function, where <math>I(S;\Omega-S)</math> is the mutual information.
+==== Asymmetric ====
+A non-monotone submodular function which is not symmetric is called asymmetric.
+; Directed cuts : Let <math>\Omega=\{v_1,v_2,\dots,v_n\}</math> be the vertices of a [[directed graph]]. For any set of vertices <math>S\subseteq \Omega</math> let <math>f(S)</math> denote the number of edges <math>e=(u,v)</math> such that <math>u\in S</math> and <math>v\in \Omega-S</math>. This can be generalized by adding non-negative weights to the directed edges.
+== Continuous extensions ==
+=== Lovász extension ===
+This extension is named after mathematician [[László Lovász]]. Consider any vector <math>\bold{x}=\{x_1,x_2,\dots,x_n\}</math> such that each <math>0\leq x_i\leq 1</math>. Then the Lovász extension is defined as <math>f^L(\bold{x})=\mathbb{E}(f(\{i|x_i\geq \lambda\}))</math> where the expectation is over <math>\lambda</math> chosen from the [[uniform distribution (continuous)|uniform distribution]] on the interval <math>[0,1]</math>. The Lovász extension is a convex function.
+=== Multilinear extension ===
+Consider any vector <math>\bold{x}=\{x_1,x_2,\ldots,x_n\}</math> such that each <math>0\leq x_i\leq 1</math>. Then the multilinear extension is defined as <math>F(\bold{x})=\sum_{S\subseteq \Omega} f(S) \prod_{i\in S} x_i \prod_{i\notin S} (1-x_i)</math>.
+=== Convex closure ===
+Consider any vector <math>\bold{x}=\{x_1,x_2,\dots,x_n\}</math> such that each <math>0\leq x_i\leq 1</math>. Then the convex closure is defined as <math>f^-(\bold{x})=\min(\sum_S \alpha_S f(S):\sum_S \alpha_S 1_S=\bold{x},\sum_S \alpha_S=1,\alpha_S\geq 0)</math>. It can be shown that <math>f^L(\bold{x})=f^-(\bold{x})</math>.
+=== Concave closure ===
+Consider any vector <math>\bold{x}=\{x_1,x_2,\dots,x_n\}</math> such that each <math>0\leq x_i\leq 1</math>. Then the concave closure is defined as <math>f^+(\bold{x})=\max(\sum_S \alpha_S f(S):\sum_S \alpha_S 1_S=\bold{x},\sum_S \alpha_S=1,\alpha_S\geq 0)</math>.
+== Properties ==
+# The class of submodular functions is [[closure (mathematics)|closed]] under non-negative [[linear combination]]s. Consider any submodular function <math>f_1,f_2,\ldots,f_k</math> and non-negative numbers <math>\alpha_1,\alpha_2,\ldots,\alpha_k</math>. Then the function <math>g</math> defined by <math>g(S)=\sum_{i=1}^k \alpha_i f_i(S)</math> is submodular. Furthermore, for any submodular function <math>f</math>, the function defined by <math>g(S)=f(\Omega \setminus S)</math> is submodular. The function <math>g(S)=\min(f(S),c)</math>, where <math>c</math> is a real number, is submodular whenever <math>f</math> is monotonic.
+# If <math>f:2^\Omega\rightarrow \mathbb{R}_+</math> is a submodular function then <math>g:2^\Omega\rightarrow \mathbb{R}_+</math> defined as <math>g(S)=\phi(f(S))</math> where <math>\phi</math> is a [[concave]] function, is also a submodular function.
+# Consider a random process where a set <math>T</math> is chosen with each element in <math>\Omega</math> being included in <math>T</math> independently with probability <math>p</math>. Then the following inequality is true <math>\mathbb{E}[f(T)]\geq p f(\Omega)+(1-p) f(\varnothing)</math> where <math>\varnothing</math> is the empty set. More generally consider the following random process where a set <math>S</math> is constructed as follows. For each of <math>1\leq i\leq l, A_i\subseteq \Omega</math> construct <math>S_i</math> by including each element in <math>A_i</math> independently into <math>S_i</math> with probability <math>p_i</math>. Furthermore let <math>S=\cup_{i=1}^l S_i</math>. Then the following inequality is true <math>\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)</math>.{{Citation needed|date=November 2013}}
+== Optimization problems ==
+Submodular functions have properties which are very similar to [[convex function|convex]] and [[concave function]]s. 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 <math>S\subseteq \Omega</math> which minimizes a submodular function subject to no constraints. This problem is computable in (strongly)<ref name="IFF" /><ref name="Schrijver" /> [[polynomial time]].<ref name="GLS" /><ref name="Cunningham" /> 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 algorithm]]s or [[local search (optimization)|local search algorithm]]s. The problem of maximizing a symmetric non-monotone submodular function subject to no constraints admits a 1/2 approximation algorithm.<ref name="FMV" /> 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.<ref name="BFNS" /> The problem of maximizing a monotone submodular function subject to a cardinality constraint admits a <math>1 - 1/e</math> approximation algorithm.<ref name="NVF" /> 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 <math>1 - 1/e</math> approximation algorithm.<ref name="CCPV" /><ref name="FW" />
+== See also ==
+* [[Supermodular function]]
+* [[Polymatroid]]
+* [[Matroid]]
+== Citations ==
+{{reflist|
+refs=
+<ref name="GLS">M. Gr&ouml;tschel, [[László Lovász|L. Lovasz]] and [[Alexander Schrijver|A. Schrijver]], The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981), pp. 169–197.</ref>
+<ref name="Cunningham">W. H. Cunningham, On submodular function minimization,  Combinatorica,5 (1985),pp. 185–192.</ref>
+<ref name="IFF">S. Iwata, L. Fleischer, and S. Fujishige, A combinatorial strongly polynomial algorithm for minimizing submodular functions, J. ACM 48 (2001), pp. 761–777.</ref>
+<ref name="Schrijver">[[Alexander Schrijver|A. Schrijver]], A combinatorial algorithm minimizing submodular functions in strongly polynomial time, J. Combin. Theory Ser. B 80 (2000), pp. 346–355.</ref>
+<ref name="FMV">[[Uriel Feige|U. Feige]], V. Mirrokni and J. Vondr&aacute;k, Maximizing non-monotone submodular functions, Proc. of 48th FOCS (2007), pp. 461–471.</ref>
+<ref name="NVF">[[George Nemhauser|G. L. Nemhauser]], L. A. Wolsey and M. L. Fisher, An analysis of approximations for maximizing submodular set functions I, Mathematical Programming 14 (1978), 265–294.</ref>
+<ref name="CCPV">G. Calinescu, C. Chekuri, M. P&aacute;l and J. Vondr&aacute;k, Maximizing a submodular set function subject to a matroid constraint, SIAM J. Comp. 40:6 (2011), 1740-1766.</ref>
+<ref name="BFNS">N. Buchbinder, M. Feldman, J. Naor and R. Schwartz, A tight linear time (1/2)-approximation for unconstrained submodular maximization, Proc. of 53rd FOCS (2012), pp. 649-658.</ref>
+<ref name="FW">Y. Filmus, J. Ward, A tight combinatorial algorithm for submodular maximization subject to a matroid constraint, Proc. of 53rd FOCS (2012), pp. 659-668.</ref>
+}}
+== References ==
+*{{Citation|last=Schrijver|first=Alexander|authorlink=Alexander Schrijver|year=2003|title=Combinatorial Optimization|location=|publisher=[[Springer Publishing|Springer]]|isbn=3-540-44389-4}}
+*{{Citation|last=Lee|first=Jon|authorlink=Jon Lee (mathematician)|year= 2004 |title=A First Course in Combinatorial Optimization |location=|publisher=[[Cambridge University Press]]|isbn= 0-521-01012-8}}
+*{{Citation|last=Fujishige|first=Satoru|year=2005|title=Submodular Functions and Optimization|location=|publisher=[[Elsevier]]|isbn=0-444-52086-4}}
+*{{Citation|last=Narayanan|first=H.|year= 1997 |title=Submodular Functions and Electrical Networks|location=|publisher=|isbn= 0-444-82523-1}}
+*{{citation | last=Oxley | first=James G. | title=Matroid theory | series=Oxford Science Publications | location=Oxford | publisher=[[Oxford University Press]] | year=1992 | isbn=0-19-853563-5 | zbl=0784.05002 }}
+<!--- Categories --->
+[[Category:Combinatorial optimization| ]]
+[[Category:Approximation algorithms| ]]
+[[Category:Matroid theory|Matroid theory]]

Brodal queue: Difference between revisions

Latest revision as of 04:33, 4 October 2013

Contents

Definition

Types of submodular functions

Monotone

Non-monotone

Symmetric

Asymmetric

Continuous extensions

Lovász extension

Multilinear extension

Convex closure

Concave closure

Properties

Optimization problems

See also

Citations

References

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Brodal queue: Difference between revisions

Latest revision as of 04:33, 4 October 2013

Definition

Types of submodular functions

Monotone

Non-monotone

Symmetric

Asymmetric

Continuous extensions

Lovász extension

Multilinear extension

Convex closure

Concave closure

Properties

Optimization problems

See also

Citations

References

Navigation menu

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