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### EMIS 8374: Network Flows

“Easy” Integer Programming Problems: Network Flow Problems

updated 4 April 2004

Basic Feasible Solutions

Standard Form

Matrix Representation

Observation: The last row of the matrix is equal to –1 times the sum of the other rows.

MCNF LPs always have one redundant row.

Matrix Representation without the constraint for node 6

A BFS: B = {x12, x13, x24, x35, x56}

Solving for the BFS

Constraints after non-basic variables are removed:

Solution: x24 = 0, x12 = 0, x13 = 1, x35 = 1, x56 = 1

Kramer’s (a.k.a Cramer’s) Rule

Component j of x = A-1b is

Take the matrix A and replace

column j with the vector b.

Total Unimodularity

- A square, integer matrix is unimodular if its determinant is 1 or -1.
- An integer matrix A is called totally unimodular (TU) if every square, nonsingular submatrix of A is unimodular.
- From Cramer’s rule, it follows that if A is TU and b is an integer vector, then every BFS of the constraint system Ax = b is integer.
- Examples:
- The matrix AB from the shortest path example is TU.
- The matrix A from the shortest path example is TU.
- The constraint matrix for any MCNF LP is TU.

TU Theorem

- An integer matrix A is TU 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 matrix A is TU if and only if is AT TU.
- The matrix A is TU if and only if [A, I] is TU. Where I is the identity matrix.

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