“Easy” Integer Programming Problems: Network Flow Problems EMIS 8374: Network Flows “Easy” Integer Programming Problems: Network Flow Problems updated 4 April 2004

Basic Feasible Solutions Standard Form

Basic Feasible Solutions

Vector-Matrix Representation

LP Formulation of Shortest Path Example

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

Solving for the BFS with Matrix Algebra

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 Examples: 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.

MCNF LPs are TU Flow Balance: A is TU, so AT is TU. Capacity