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.

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.

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.

Node-Arc Incidence Matrices are TUM 1 2 3

MCNFP LP

Constraint Matrix for Example MCNFP in Standard Form

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.