Mixing
On Matrices that are close to Total Unimodularity
$begingroup$
Assume we have some matrix $A$ which is totally unimodular(TU). By a well known results we have that the polyhedron $P = {Axle b}$ has integer vertices for all $b in mathbb{Z}^n$. My question is about matrices which are in a way close to being TU. For example assume that you create a matrix $A'$ by changing a single entry of $A$ which then violates TU, but only in a minimal sense, e.g. you can find only a small linear number of submatrices which have determinant not in ${-1,0,1}$. I am mainly wondering about the following aspects:
- For which vectors $b in mathbb{Z}^n$ is ${A'xle b}$ still integral?
- Can we make general statements about which vertices are affected by the change and which might remain integral?
- Is there additional structure that allows more general statements? For example what if $A' = [B C]$ with both $B,C$ being TU?
For 2. it is clear that vertices for which the active constraints are unaffected are still integral, but to me it is not clear how to determine which constraints exactly are affected by such a change or which are affected by violating submatrices.
I am grateful for all helpful suggestions or literature recommendations, as it seems to be very hard to find information/literature about these kinds of problems.
linear-programming polyhedra total-unimodularity
asked Jan 11 at 9:02
HajakuHajaku
12
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add a comment |
$begingroup$
Assume we have some matrix $A$ which is totally unimodular(TU). By a well known results we have that the polyhedron $P = {Axle b}$ has integer vertices for all $b in mathbb{Z}^n$. My question is about matrices which are in a way close to being TU. For example assume that you create a matrix $A'$ by changing a single entry of $A$ which then violates TU, but only in a minimal sense, e.g. you can find only a small linear number of submatrices which have determinant not in ${-1,0,1}$. I am mainly wondering about the following aspects:
- For which vectors $b in mathbb{Z}^n$ is ${A'xle b}$ still integral?
- Can we make general statements about which vertices are affected by the change and which might remain integral?
- Is there additional structure that allows more general statements? For example what if $A' = [B C]$ with both $B,C$ being TU?
For 2. it is clear that vertices for which the active constraints are unaffected are still integral, but to me it is not clear how to determine which constraints exactly are affected by such a change or which are affected by violating submatrices.
I am grateful for all helpful suggestions or literature recommendations, as it seems to be very hard to find information/literature about these kinds of problems.
linear-programming polyhedra total-unimodularity
asked Jan 11 at 9:02
HajakuHajaku
12
$endgroup$
add a comment |
$begingroup$
Assume we have some matrix $A$ which is totally unimodular(TU). By a well known results we have that the polyhedron $P = {Axle b}$ has integer vertices for all $b in mathbb{Z}^n$. My question is about matrices which are in a way close to being TU. For example assume that you create a matrix $A'$ by changing a single entry of $A$ which then violates TU, but only in a minimal sense, e.g. you can find only a small linear number of submatrices which have determinant not in ${-1,0,1}$. I am mainly wondering about the following aspects:
- For which vectors $b in mathbb{Z}^n$ is ${A'xle b}$ still integral?
- Can we make general statements about which vertices are affected by the change and which might remain integral?
- Is there additional structure that allows more general statements? For example what if $A' = [B C]$ with both $B,C$ being TU?
For 2. it is clear that vertices for which the active constraints are unaffected are still integral, but to me it is not clear how to determine which constraints exactly are affected by such a change or which are affected by violating submatrices.
I am grateful for all helpful suggestions or literature recommendations, as it seems to be very hard to find information/literature about these kinds of problems.
linear-programming polyhedra total-unimodularity
asked Jan 11 at 9:02
HajakuHajaku
12
$endgroup$
Assume we have some matrix $A$ which is totally unimodular(TU). By a well known results we have that the polyhedron $P = {Axle b}$ has integer vertices for all $b in mathbb{Z}^n$. My question is about matrices which are in a way close to being TU. For example assume that you create a matrix $A'$ by changing a single entry of $A$ which then violates TU, but only in a minimal sense, e.g. you can find only a small linear number of submatrices which have determinant not in ${-1,0,1}$. I am mainly wondering about the following aspects:
- For which vectors $b in mathbb{Z}^n$ is ${A'xle b}$ still integral?
- Can we make general statements about which vertices are affected by the change and which might remain integral?
- Is there additional structure that allows more general statements? For example what if $A' = [B C]$ with both $B,C$ being TU?
For 2. it is clear that vertices for which the active constraints are unaffected are still integral, but to me it is not clear how to determine which constraints exactly are affected by such a change or which are affected by violating submatrices.
I am grateful for all helpful suggestions or literature recommendations, as it seems to be very hard to find information/literature about these kinds of problems.
linear-programming polyhedra total-unimodularity
linear-programming polyhedra total-unimodularity
asked Jan 11 at 9:02
HajakuHajaku
12
asked Jan 11 at 9:02
HajakuHajaku
12
asked Jan 11 at 9:02
HajakuHajaku
12
asked Jan 11 at 9:02
HajakuHajaku
12
asked Jan 11 at 9:02
HajakuHajaku
12
12
add a comment |
add a comment |
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