1. Kramer’s (a.k.a Cramer’s) Rule
Component j of
x = A-1b is
Form Bj by replacing column j of A with b.

2. Total Unimodularity
A square, integer matrix B is unimodular (UM) if its determinant is 1 or -1.
An integer matrix A is called totally unimodular (TUM) if every square, nonsingular submatrix of A is UM.
From Cramer’s rule, it follows that if A is TUM and b is an integer vector, then every BFS of the constraint system Ax = b is integer.

3. TUM Theorem An integer matrix A is TUM if
All entries are -1, 0 or 1
At most two non-zero entries appear in any column
The rows of A can be partitioned into two disjoint sets such that
If a column has two entries of the same sign, their rows are in different sets.
If a column has two entries of different signs, their rows are in the same set.
The MCNFP constraint matrices are TUM.

4. Node-Arc Incidence Matrices are TUM1 2 3

5. MCNFP LP

6. Constraint Matrix for Example MCNFP in Standard Form

7. Cramer’s Rule & MCNFP The constraint matrix of an MCNFP LP is TUM.
Any BFS of the MCNFP LP is integral.
We can use the Simplex method to solve MCNFP.