Mathematics

Questions tagged [total-unimodularity]

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.

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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
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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 -1 -1 1 1 1 1 0 0 0 0 0…
remo
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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\\ \vdots & \vdots &…
Rafael Colares
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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 a…
Jill Clover
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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
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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 $(x_1,...,x_k)$, $1\leq k<n$ and obtain the new…
user91360
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How many $m \times n$ totally unimodular matrices are there?

Let $A \in M_{m\times n}(\mathbb{Z})$ , then $A$ is totally unimodular iff for any $B$ , square submatrix of $A$ , then $|\det(B)| \leq 1$ . If $A$ is itself a square matrix, then we need also $|\det(A)|=1$ . My question will be the…
Tiziano Brunelli
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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
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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
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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
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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 $$P…
UniStuffz
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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.
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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 & 0…
KRH
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Total unimodularity towards linear programming relaxation

Currently, I am studying total unimodularity. I was reading this link: https://ostad.nit.ac.ir/upload/Integer_Programming_1.pdf , from page 38-41 and I came across the statement: It is clear that when matrix A is totally unimodular, the linear…
Michelle Gunawan
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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
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