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Linear Programs with Totally Unimodular Matrices

Linear Programs with Totally Unimodular Matrices. updated 21 April2008. Basic Feasible Solutions. Standard Form. Basic Feasible Solutions. Vector-Matrix Representation. 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.

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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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