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Linear Programs with Totally Unimodular Matrices

This updated guide explores the use of totally unimodular matrices in linear programming, focusing on basic feasible solutions through examples and matrix representations. Learn how to apply Cramer's Rule, understand total unimodularity, and solve for variables within the constraints. Discover sufficient conditions for total unimodularity and explore the matrix of flow balance constraints. Utilize expansion by minors and theorems related to totally unimodular matrices. Enhance your LP solving skills with this comprehensive resource.

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Linear Programs with Totally Unimodular Matrices

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  1. Linear Programs with Totally Unimodular Matrices updated 21 April2008

  2. Basic Feasible Solutions Standard Form

  3. Basic Feasible Solutions

  4. Vector-Matrix Representation

  5. Example MCNFP -2 (3, 2,5) (4, 1,3) 2 5 -3 1 4 (1, 0,2) (4, 0,3) (2, 0,2) 3 0

  6. LP for Example MCNFP Min 3x12 + 2 x13 + x23 + 4 x24 + 4 x34 s.t. x12 + x13 = 5 {Node 1} x23 + x24 – x12 = -2 {Node 2} x34 – x13 - x23 = 0 {Node 3} – x24 - x34 = -3 {Node 4} 2  x12 5, 0  x13 2, 0  x23 2, 1  x24 3, 0  x34 3,

  7. Matrix Representation of Flow Balance Constraints

  8. Solving for a Basic Feasible Solution

  9. Cramer’s Rule Use determinants to solve x=A-1b. Take the matrix A and replace column j with the vector b to form matrix Bj.

  10. Using Cramer’s Rule to Solve for x12

  11. 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.

  12. 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.

  13. Sufficient Conditions for 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 M1 and M2 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.

  14. The Matrix of Flow Balance Constraints • Every column has exactly one +1 and exactly one -1. • This satisfies conditions 1 and 2. • Let the row partition be M1 = {all rows} and M2 = {}. • This satisfies condition 3. • Thus the flow balance constraint matrix is TU.

  15. Using Cramer’s Rule to Solve for x12

  16. Expansion by Minors: 4-by-4 Matrix

  17. Expansion by Minors: 3-by-3 Matrix

  18. Using Cramer’s Rule to Solve for x12

  19. Using Cramer’s Rule to Solve for x12 • When we expand along minors, the determinants of the submatrices will be +1, -1, or 0. • Therefore, the determinant will be an integer: (5)(+1, -1, or 0) + (-2) (+1, -1, or 0) + 0 + (-3) (+1, -1, or 0).

  20. Using Cramer’s Rule to Solve for x12

  21. TU Theorems • Matrix A is TU if and only if AT is TU. • Matrix A is TU if and only if [A, I] is TU. • I is the identity matrix. • If the constraint matrix for an IP is TU, then its LP relaxation has an integral optimal solution. • The BFSs of an MCNF LP are integer valued.

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