A totally unimodular matrix is a matrix for which every square non-singular submatrix is unimodular.
A totally unimodular matrix is a matrix for which every square non-singular submatrix is unimodular. For more information see here.
Questions tagged [total-unimodularity]
70 questions
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votes
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Prove that the matrix is totally unimodular
Is there any (theoretic) way I can prove the matrix is totally unimodular? I have tested it by Matlab and know it is TU, however I cannot prove it.
-1 -1 -1 -1 0 0 0 0 0 0 0 0
0 0 0 0 -1 -1 -1 -1 0 0 0 0
0 0 0 0 0 0 0 0 -1 -1…
remo
- 441
5
votes
1 answer
Is this block matrix totally unimodular?
Suppose matrix $A \in \mathbb{R}^{m×n}$ has the consecutive ones property and, thus, is totally unimodular. Is the following block matrix also totally unimodular (TU)?
$$B = \begin{pmatrix}
A & 0 & \dots & 0\\
0 & A & \dots & 0\\
…
Rafael Colares
- 178
5
votes
1 answer
How to prove in a graph $G$, its incidence matrix $A$ is totally unimodular if and only if $G$ is a bipartite graph?
I'm learning network and transportation model. The question is not from my homework. I'm just curious about:
In a graph $G$, its incidence matrix $A$ is totally unimodular if and only if $G$ is a bipartite graph.
Could anyone good at proving give…
Jill Clover
- 4,907
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- 33
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5
votes
2 answers
Is this block matrix also totally unimodular?
Suppose matrix $A\in \mathbb R^{m\times n}$ is totally unimodular (TUM). Is the following matrix also TUM?
$$
\begin{pmatrix}
A & 0 & 0 \\
0 & A & 0 \\
0 & 0 & A\\
I & I & I\\
\end{pmatrix}
$$
Thanks.
user91360
- 1,231
4
votes
1 answer
How to prove that incidence matrix is totally unimodular
Problem: Prove that the incidence matrix $A$ of a graph define as follows: $$A_{ij}= \begin{cases}
-1 & \text{ if the edge } e_j \text{ leaves the vertex } v_i\\ 1 & \text{ if the edge } e_j \text{ enters the vertex } v_i\\ 0 & \text{
otherwise}…
Tung Nguyen
- 1,278
4
votes
0 answers
Fourier Motzkin Elimination and totally unimodularity
Suppose $A\in \mathbb R^{m\times n}$ and $b\in \mathbb R^m$, and $A$ is totally unimodular (TUM).
For the system $$Ax\leq b,$$
suppose I use Fourier-Motzkin elimination to eliminate first $k$ variables $(x_1,...,x_k)$, $1\leq k
user91360
- 1,231
3
votes
1 answer
Is there a theorem that states that Integer Linear Problems with a Totally Unimodular constraint matrix are solvable in polynomial time?
Is there a theorem that states that Integer Linear Problems with a Totally Unimodular constraint matrix are solvable in polynomial time?
If the answer is positive, is it also valid for Mixed-Integer Linear Problems? And in which books can I find…
Mathgineer94
- 61
- 4
3
votes
2 answers
Pivoting fails to preserve total unimodularity
It is well known that a $\{0, \pm 1\}$ matrix $A$ is totally unimodular (TUM) if and only if matrix $A'$ obtained from $A$ by pivoting operation is totally unimodular.
Here pivoting an element $a_{ij}\neq 0$ is defined as first multiplying the row…
sam
- 131
3
votes
1 answer
General solution to linear program?
Source of the Problem
The problem comes from an application in economics concerning trade between agents to maximize aggregate "wealth". More exactly, there are $m$ agents and $n$ groups and we perform independent trials where an agent interacting…
SimonSimon
- 149
3
votes
1 answer
Adapting a proof to show that a polyhedron has integer extreme points
Prove that the polyhedron $$P =\{(x_1, \ldots , x_m, x_{m+1}): 0\leq x_m \leq 1, 0 \leq x_i \leq x_{m+1} \text{ for } i = 1, \ldots , m \}$$ has integer extreme points.
My Attempt:
I have a similar proof in my notes to prove that the polyhedron…
UniStuffz
- 932
3
votes
1 answer
Proof that the incidence matrix of a laminar family is TU.
A friend and I wrote a proof for this using the consecutive ones property that I haven't seen anywhere, so I thought I would share it here.
$\textit{Def:}$ A matrix has the Total-Unimodularity (TU) property if the determinant of all of its square…
Logan S.
- 405
2
votes
1 answer
Is the matrix by appending all 1 row to the transport coefficient matrix still totally unimodular?
It is already known that the transport flow coefficient matrix (4 sources to 3 targets in this particular example)
$$
\begin{bmatrix}
1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 &…
KRH
- 23
2
votes
2 answers
Is the matrix totally unimodular?
Let $A=\begin{bmatrix}1&0&1&1&0&0&1&0&0\\0&1&0&1&0&0&0&0&0& \\0&0&1&1&1&0&0&0&0\\0&0&-1&-1&0&1&-1&0&0\\0&0&0&-1&0&0&0&1&0\\0&0&-1&-1&0&0&0&0&1\end{bmatrix}$, is the matrix totally unimodular?
I realised that if I change the order of columns in A I…
Alex
- 147
2
votes
2 answers
A totally unimodular matrix
Is there a way to test whether the following rank-$6$ matrix is totally unimodular?
$$\begin{bmatrix}1& 0& -1& 0& 0& 0& 0& 0\\0& 0& 0& 0& 1& 0& -1& 0\\1& 0& 0& 0& 1& 0& 0& 0\\0& 1& 0& 0& 0& -1& 0& 0\\0& 0& 0& 1& 0& 0& 0& -1\\0& 1& 0& 1& 0& 0& 0&…
Turbo
- 6,319
2
votes
1 answer
LP relaxation solves the integer program but the constraint matrix is not totally unimodular
I am solving an integer program (IP) whose constraint matrix is not totally unimodular (TU). The linear programming (LP) relaxation and the original IP always have the same optimal solution, or the LP relaxation is always optimal — in the hundreds…
bissi
- 64