Mixing
graph theory: minimize the panelty
$begingroup$
Suppose that there are $M$ left nodes, and $N$ right nodes. Suppose that the $n$th right node can connect $K_n$ left nodes, $n=1,cdots,N$. Let $Lambda_m$ denote the set of right nodes that is connected to the $m$th left node, $m=1,cdots,M$. If the $m$th left node is connected to the $n$th right node, i.e., $nin Lambda_m$, then there is no penalty; otherwise, i.e., $nnotin Lambda_m$, there will be a panelty $K_n$ to the $m$th left node for not connecting to the $n$th right node, i.e., the capacity of the $n$th right node.
Then, given any assignment $Lambda_m$'s, the total panelty of the $m$th left node for not connecting the right nodes is
begin{align}
P_m=sumlimits_{nnotin Lambda_m} K_n, ~~~ m=1,cdots,M.
end{align}
The problem is to find the optimal assignment $Lambda_m$'s to minimize the maximum penalty among all the $M$ left nodes, i.e.,
begin{align}
{rm minimize} ~ max(P_1,cdots,P_M),
end{align}suppose that each right node $n$ can only connect $K_n$ left nodes.
Is there any efficient way to solve this problem?
graph-theory combinations
asked Jan 3 at 8:01
LiangLiang
312
$endgroup$
add a comment |
$begingroup$
Suppose that there are $M$ left nodes, and $N$ right nodes. Suppose that the $n$th right node can connect $K_n$ left nodes, $n=1,cdots,N$. Let $Lambda_m$ denote the set of right nodes that is connected to the $m$th left node, $m=1,cdots,M$. If the $m$th left node is connected to the $n$th right node, i.e., $nin Lambda_m$, then there is no penalty; otherwise, i.e., $nnotin Lambda_m$, there will be a panelty $K_n$ to the $m$th left node for not connecting to the $n$th right node, i.e., the capacity of the $n$th right node.
Then, given any assignment $Lambda_m$'s, the total panelty of the $m$th left node for not connecting the right nodes is
begin{align}
P_m=sumlimits_{nnotin Lambda_m} K_n, ~~~ m=1,cdots,M.
end{align}
The problem is to find the optimal assignment $Lambda_m$'s to minimize the maximum penalty among all the $M$ left nodes, i.e.,
begin{align}
{rm minimize} ~ max(P_1,cdots,P_M),
end{align}suppose that each right node $n$ can only connect $K_n$ left nodes.
Is there any efficient way to solve this problem?
graph-theory combinations
asked Jan 3 at 8:01
LiangLiang
312
$endgroup$
add a comment |
$begingroup$
Suppose that there are $M$ left nodes, and $N$ right nodes. Suppose that the $n$th right node can connect $K_n$ left nodes, $n=1,cdots,N$. Let $Lambda_m$ denote the set of right nodes that is connected to the $m$th left node, $m=1,cdots,M$. If the $m$th left node is connected to the $n$th right node, i.e., $nin Lambda_m$, then there is no penalty; otherwise, i.e., $nnotin Lambda_m$, there will be a panelty $K_n$ to the $m$th left node for not connecting to the $n$th right node, i.e., the capacity of the $n$th right node.
Then, given any assignment $Lambda_m$'s, the total panelty of the $m$th left node for not connecting the right nodes is
begin{align}
P_m=sumlimits_{nnotin Lambda_m} K_n, ~~~ m=1,cdots,M.
end{align}
The problem is to find the optimal assignment $Lambda_m$'s to minimize the maximum penalty among all the $M$ left nodes, i.e.,
begin{align}
{rm minimize} ~ max(P_1,cdots,P_M),
end{align}suppose that each right node $n$ can only connect $K_n$ left nodes.
Is there any efficient way to solve this problem?
graph-theory combinations
asked Jan 3 at 8:01
LiangLiang
312
$endgroup$
Suppose that there are $M$ left nodes, and $N$ right nodes. Suppose that the $n$th right node can connect $K_n$ left nodes, $n=1,cdots,N$. Let $Lambda_m$ denote the set of right nodes that is connected to the $m$th left node, $m=1,cdots,M$. If the $m$th left node is connected to the $n$th right node, i.e., $nin Lambda_m$, then there is no penalty; otherwise, i.e., $nnotin Lambda_m$, there will be a panelty $K_n$ to the $m$th left node for not connecting to the $n$th right node, i.e., the capacity of the $n$th right node.
Then, given any assignment $Lambda_m$'s, the total panelty of the $m$th left node for not connecting the right nodes is
begin{align}
P_m=sumlimits_{nnotin Lambda_m} K_n, ~~~ m=1,cdots,M.
end{align}
The problem is to find the optimal assignment $Lambda_m$'s to minimize the maximum penalty among all the $M$ left nodes, i.e.,
begin{align}
{rm minimize} ~ max(P_1,cdots,P_M),
end{align}suppose that each right node $n$ can only connect $K_n$ left nodes.
Is there any efficient way to solve this problem?
graph-theory combinations
graph-theory combinations
asked Jan 3 at 8:01
LiangLiang
312
asked Jan 3 at 8:01
LiangLiang
312
asked Jan 3 at 8:01
LiangLiang
312
asked Jan 3 at 8:01
LiangLiang
312
asked Jan 3 at 8:01
LiangLiang
312
312
add a comment |
add a comment |
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