1 / 13

# EMIS 8374: Network Flows - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' EMIS 8374: Network Flows' - tayte

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### EMIS 8374: Network Flows

“Easy” Integer Programming Problems: Network Flow Problems

updated 4 April 2004

Standard Form

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.

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

Constraints after non-basic variables are removed:

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

Component j of x = A-1b is

Take the matrix A and replace

column j with the vector b.

• 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.

• 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.

Flow Balance:

A is TU, so AT is TU.

Capacity