A more general case of assignment problem
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Recently I've learned hungarian algorithm for solving the assignment problem, Now I'm curious about how to solve more general problem: for given $n times m$ table select several numbers, maximizing their sum with following constraints: in each row the number of selected numbers is not less than $R_{min}$ and not more than $R_{max}$, and for each column the number of selected numbers is not less than $C_{min}$ and not more than $C_{max}$.
If $R_{min} = R_{max} = 1$ and $C_{min} = C_{max} = 1$ this is the standard assignment problem. But what can we say about more general case? Any ideas about this (maybe, at least in case $R_{min} = R_{max} = R$, $C_{min} = C_{max} = C$)? Any thoughts would be greatly appreciated.
optimization algorithms linear-programming
asked May 13 '16 at 23:14
SwistackSwistack
425211
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add a comment |
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Recently I've learned hungarian algorithm for solving the assignment problem, Now I'm curious about how to solve more general problem: for given $n times m$ table select several numbers, maximizing their sum with following constraints: in each row the number of selected numbers is not less than $R_{min}$ and not more than $R_{max}$, and for each column the number of selected numbers is not less than $C_{min}$ and not more than $C_{max}$.
If $R_{min} = R_{max} = 1$ and $C_{min} = C_{max} = 1$ this is the standard assignment problem. But what can we say about more general case? Any ideas about this (maybe, at least in case $R_{min} = R_{max} = R$, $C_{min} = C_{max} = C$)? Any thoughts would be greatly appreciated.
optimization algorithms linear-programming
asked May 13 '16 at 23:14
SwistackSwistack
425211
$endgroup$
add a comment |
$begingroup$
Recently I've learned hungarian algorithm for solving the assignment problem, Now I'm curious about how to solve more general problem: for given $n times m$ table select several numbers, maximizing their sum with following constraints: in each row the number of selected numbers is not less than $R_{min}$ and not more than $R_{max}$, and for each column the number of selected numbers is not less than $C_{min}$ and not more than $C_{max}$.
If $R_{min} = R_{max} = 1$ and $C_{min} = C_{max} = 1$ this is the standard assignment problem. But what can we say about more general case? Any ideas about this (maybe, at least in case $R_{min} = R_{max} = R$, $C_{min} = C_{max} = C$)? Any thoughts would be greatly appreciated.
optimization algorithms linear-programming
asked May 13 '16 at 23:14
SwistackSwistack
425211
$endgroup$
Recently I've learned hungarian algorithm for solving the assignment problem, Now I'm curious about how to solve more general problem: for given $n times m$ table select several numbers, maximizing their sum with following constraints: in each row the number of selected numbers is not less than $R_{min}$ and not more than $R_{max}$, and for each column the number of selected numbers is not less than $C_{min}$ and not more than $C_{max}$.
If $R_{min} = R_{max} = 1$ and $C_{min} = C_{max} = 1$ this is the standard assignment problem. But what can we say about more general case? Any ideas about this (maybe, at least in case $R_{min} = R_{max} = R$, $C_{min} = C_{max} = C$)? Any thoughts would be greatly appreciated.
optimization algorithms linear-programming
optimization algorithms linear-programming
asked May 13 '16 at 23:14
SwistackSwistack
425211
asked May 13 '16 at 23:14
SwistackSwistack
425211
asked May 13 '16 at 23:14
SwistackSwistack
425211
asked May 13 '16 at 23:14
SwistackSwistack
425211
asked May 13 '16 at 23:14
SwistackSwistack
425211
425211
add a comment |
add a comment |
1 Answer
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The general case can be modeled as a flow problem in a bipartite graph $G=(V_1cup V_2,E)$, with $|V_1|=n$ and $|V_2|=m$.
To take into account your constraints, add a source $s$ and link to all vertices of $V_1$, and a sink $t$, linked to all vertices of $V_2$. Impose that the entering flow in each node of $V_1$ is at least $R_{min}$ and at most $R_{max}$, and that the exiting flow from each node of $V_2$ is at least $C_{min}$ and at most $C_{max}$, and you are done.
answered May 14 '16 at 2:22
KuifjeKuifje
7,2722726
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1 Answer
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1 Answer
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$begingroup$
The general case can be modeled as a flow problem in a bipartite graph $G=(V_1cup V_2,E)$, with $|V_1|=n$ and $|V_2|=m$.
To take into account your constraints, add a source $s$ and link to all vertices of $V_1$, and a sink $t$, linked to all vertices of $V_2$. Impose that the entering flow in each node of $V_1$ is at least $R_{min}$ and at most $R_{max}$, and that the exiting flow from each node of $V_2$ is at least $C_{min}$ and at most $C_{max}$, and you are done.
answered May 14 '16 at 2:22
KuifjeKuifje
7,2722726
$endgroup$
add a comment |
$begingroup$
The general case can be modeled as a flow problem in a bipartite graph $G=(V_1cup V_2,E)$, with $|V_1|=n$ and $|V_2|=m$.
To take into account your constraints, add a source $s$ and link to all vertices of $V_1$, and a sink $t$, linked to all vertices of $V_2$. Impose that the entering flow in each node of $V_1$ is at least $R_{min}$ and at most $R_{max}$, and that the exiting flow from each node of $V_2$ is at least $C_{min}$ and at most $C_{max}$, and you are done.
answered May 14 '16 at 2:22
KuifjeKuifje
7,2722726
$endgroup$
add a comment |
$begingroup$
The general case can be modeled as a flow problem in a bipartite graph $G=(V_1cup V_2,E)$, with $|V_1|=n$ and $|V_2|=m$.
To take into account your constraints, add a source $s$ and link to all vertices of $V_1$, and a sink $t$, linked to all vertices of $V_2$. Impose that the entering flow in each node of $V_1$ is at least $R_{min}$ and at most $R_{max}$, and that the exiting flow from each node of $V_2$ is at least $C_{min}$ and at most $C_{max}$, and you are done.
answered May 14 '16 at 2:22
KuifjeKuifje
7,2722726
$endgroup$
The general case can be modeled as a flow problem in a bipartite graph $G=(V_1cup V_2,E)$, with $|V_1|=n$ and $|V_2|=m$.
To take into account your constraints, add a source $s$ and link to all vertices of $V_1$, and a sink $t$, linked to all vertices of $V_2$. Impose that the entering flow in each node of $V_1$ is at least $R_{min}$ and at most $R_{max}$, and that the exiting flow from each node of $V_2$ is at least $C_{min}$ and at most $C_{max}$, and you are done.
answered May 14 '16 at 2:22
KuifjeKuifje
7,2722726
answered May 14 '16 at 2:22
KuifjeKuifje
7,2722726
answered May 14 '16 at 2:22
KuifjeKuifje
7,2722726
answered May 14 '16 at 2:22
KuifjeKuifje
7,2722726
7,2722726
add a comment |
add a comment |
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