Maximum weight matching with repeated nodes
We are given two sets of nodes $A$ and $B$ forming a graph where each element $x in A$ can be connected with an element $y in B$ with different possible weights. The graph can be explained in two ways.
There are multiple edges possible between elements of $X$ and $Y$ having different weights.
Elements of $A$ can be matched with repeated nodes of elements of $B$ with different edge weights.
Let $E(x)$ be the set of edges involving element x. Then the maximum weight matching problem can be formulated as follows:
Let $x_{ij}$ takes value 1, if edge $(i, j) in E$ is selected and $w_{ij}$ be weight of edge $(i, j) in E$.
$$text{minimize} sum_{(i, j) in E} x_{ij}w_{ij}$$
$$sum_{j in E(i)}x_{ij} le 1 forall i in A$$
$$sum_{i in E(j)}x_{ij} le 1 forall j in B$$
$$x_{ij} = {0, 1} quad forall (i, j) in E$$
My question is whether this problem satisfy the total unimodularity condition. If it does, how can we prove it? I know proof for the standard problem where you consider submatrices and find the determinant.
Can we still use network flow algorithms to solve this problem?
optimization discrete-optimization bipartite-graph matching-theory
asked Jan 4 at 5:30
DanDan
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We are given two sets of nodes $A$ and $B$ forming a graph where each element $x in A$ can be connected with an element $y in B$ with different possible weights. The graph can be explained in two ways.
There are multiple edges possible between elements of $X$ and $Y$ having different weights.
Elements of $A$ can be matched with repeated nodes of elements of $B$ with different edge weights.
Let $E(x)$ be the set of edges involving element x. Then the maximum weight matching problem can be formulated as follows:
Let $x_{ij}$ takes value 1, if edge $(i, j) in E$ is selected and $w_{ij}$ be weight of edge $(i, j) in E$.
$$text{minimize} sum_{(i, j) in E} x_{ij}w_{ij}$$
$$sum_{j in E(i)}x_{ij} le 1 forall i in A$$
$$sum_{i in E(j)}x_{ij} le 1 forall j in B$$
$$x_{ij} = {0, 1} quad forall (i, j) in E$$
My question is whether this problem satisfy the total unimodularity condition. If it does, how can we prove it? I know proof for the standard problem where you consider submatrices and find the determinant.
Can we still use network flow algorithms to solve this problem?
optimization discrete-optimization bipartite-graph matching-theory
asked Jan 4 at 5:30
DanDan
11
New contributor
Dan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
We are given two sets of nodes $A$ and $B$ forming a graph where each element $x in A$ can be connected with an element $y in B$ with different possible weights. The graph can be explained in two ways.
There are multiple edges possible between elements of $X$ and $Y$ having different weights.
Elements of $A$ can be matched with repeated nodes of elements of $B$ with different edge weights.
Let $E(x)$ be the set of edges involving element x. Then the maximum weight matching problem can be formulated as follows:
Let $x_{ij}$ takes value 1, if edge $(i, j) in E$ is selected and $w_{ij}$ be weight of edge $(i, j) in E$.
$$text{minimize} sum_{(i, j) in E} x_{ij}w_{ij}$$
$$sum_{j in E(i)}x_{ij} le 1 forall i in A$$
$$sum_{i in E(j)}x_{ij} le 1 forall j in B$$
$$x_{ij} = {0, 1} quad forall (i, j) in E$$
My question is whether this problem satisfy the total unimodularity condition. If it does, how can we prove it? I know proof for the standard problem where you consider submatrices and find the determinant.
Can we still use network flow algorithms to solve this problem?
optimization discrete-optimization bipartite-graph matching-theory
asked Jan 4 at 5:30
DanDan
11
New contributor
Dan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
We are given two sets of nodes $A$ and $B$ forming a graph where each element $x in A$ can be connected with an element $y in B$ with different possible weights. The graph can be explained in two ways.
There are multiple edges possible between elements of $X$ and $Y$ having different weights.
Elements of $A$ can be matched with repeated nodes of elements of $B$ with different edge weights.
Let $E(x)$ be the set of edges involving element x. Then the maximum weight matching problem can be formulated as follows:
Let $x_{ij}$ takes value 1, if edge $(i, j) in E$ is selected and $w_{ij}$ be weight of edge $(i, j) in E$.
$$text{minimize} sum_{(i, j) in E} x_{ij}w_{ij}$$
$$sum_{j in E(i)}x_{ij} le 1 forall i in A$$
$$sum_{i in E(j)}x_{ij} le 1 forall j in B$$
$$x_{ij} = {0, 1} quad forall (i, j) in E$$
My question is whether this problem satisfy the total unimodularity condition. If it does, how can we prove it? I know proof for the standard problem where you consider submatrices and find the determinant.
Can we still use network flow algorithms to solve this problem?
optimization discrete-optimization bipartite-graph matching-theory
optimization discrete-optimization bipartite-graph matching-theory
asked Jan 4 at 5:30
DanDan
11
New contributor
Dan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 4 at 5:30
DanDan
11
New contributor
Dan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Jan 4 at 5:52
Dan
asked Jan 4 at 5:30
DanDan
11
New contributor
Dan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked Jan 4 at 5:30
DanDan
11
asked Jan 4 at 5:30
DanDan
11
11
New contributor
Dan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
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