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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.

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## EMIS 8374: Network Flows

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

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