Kramer

1
Kramers (a.k.a Cramers) 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 Cramers 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 TUM
5
MCNFP LP
6
Constraint Matrix for Example MCNFP in Standard
Form
7
Cramers 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.