graph theory: minimize the panelty












1












$begingroup$


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







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$endgroup$

















    1












    $begingroup$


    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







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      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







      share|cite|improve this question









      $endgroup$




      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






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 3 at 8:01









      LiangLiang

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