Property of submodular non-decreasing function
$begingroup$
Let $f:mathcal{P}(N) longrightarrow mathbb{R}$
be a set function. $f$ is submodular if
begin{align}
f(A) + f(B)
&geq
f(A cup B) + f(A cap B)
&text{for all } A, B in mathcal{P}(N),
end{align}
$f$ is non-decreasing if
begin{align}
f(A)
&leq f(B)
&text{whenever } A subseteq B.
end{align}
Show that $f$ is non-decreasing and submodular if and only if
begin{align}
f(A)
&leq
f(B) + sum_{j in A setminus B}big( f(Bcup{j}) - f(B)big)
&text{for all } A, B in mathcal{P}(N).
end{align}
Discussion. I am able to show $Longrightarrow$ and to show that $Longleftarrow$ implies $f$ is non-decreasing. But how to show that $f$ is submodular?
linear-programming integer-programming discrete-optimization
asked Dec 8 '18 at 15:52
BernoulliBernoulli
33618
$endgroup$
add a comment |
$begingroup$
Let $f:mathcal{P}(N) longrightarrow mathbb{R}$
be a set function. $f$ is submodular if
begin{align}
f(A) + f(B)
&geq
f(A cup B) + f(A cap B)
&text{for all } A, B in mathcal{P}(N),
end{align}
$f$ is non-decreasing if
begin{align}
f(A)
&leq f(B)
&text{whenever } A subseteq B.
end{align}
Show that $f$ is non-decreasing and submodular if and only if
begin{align}
f(A)
&leq
f(B) + sum_{j in A setminus B}big( f(Bcup{j}) - f(B)big)
&text{for all } A, B in mathcal{P}(N).
end{align}
Discussion. I am able to show $Longrightarrow$ and to show that $Longleftarrow$ implies $f$ is non-decreasing. But how to show that $f$ is submodular?
linear-programming integer-programming discrete-optimization
asked Dec 8 '18 at 15:52
BernoulliBernoulli
33618
$endgroup$
add a comment |
$begingroup$
Let $f:mathcal{P}(N) longrightarrow mathbb{R}$
be a set function. $f$ is submodular if
begin{align}
f(A) + f(B)
&geq
f(A cup B) + f(A cap B)
&text{for all } A, B in mathcal{P}(N),
end{align}
$f$ is non-decreasing if
begin{align}
f(A)
&leq f(B)
&text{whenever } A subseteq B.
end{align}
Show that $f$ is non-decreasing and submodular if and only if
begin{align}
f(A)
&leq
f(B) + sum_{j in A setminus B}big( f(Bcup{j}) - f(B)big)
&text{for all } A, B in mathcal{P}(N).
end{align}
Discussion. I am able to show $Longrightarrow$ and to show that $Longleftarrow$ implies $f$ is non-decreasing. But how to show that $f$ is submodular?
linear-programming integer-programming discrete-optimization
asked Dec 8 '18 at 15:52
BernoulliBernoulli
33618
$endgroup$
Let $f:mathcal{P}(N) longrightarrow mathbb{R}$
be a set function. $f$ is submodular if
begin{align}
f(A) + f(B)
&geq
f(A cup B) + f(A cap B)
&text{for all } A, B in mathcal{P}(N),
end{align}
$f$ is non-decreasing if
begin{align}
f(A)
&leq f(B)
&text{whenever } A subseteq B.
end{align}
Show that $f$ is non-decreasing and submodular if and only if
begin{align}
f(A)
&leq
f(B) + sum_{j in A setminus B}big( f(Bcup{j}) - f(B)big)
&text{for all } A, B in mathcal{P}(N).
end{align}
Discussion. I am able to show $Longrightarrow$ and to show that $Longleftarrow$ implies $f$ is non-decreasing. But how to show that $f$ is submodular?
linear-programming integer-programming discrete-optimization
linear-programming integer-programming discrete-optimization
asked Dec 8 '18 at 15:52
BernoulliBernoulli
33618
asked Dec 8 '18 at 15:52
BernoulliBernoulli
33618
asked Dec 8 '18 at 15:52
BernoulliBernoulli
33618
asked Dec 8 '18 at 15:52
BernoulliBernoulli
33618
asked Dec 8 '18 at 15:52
BernoulliBernoulli
33618
33618
add a comment |
add a comment |
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