Kramer’s (a.k.a Cramer’s) Rule

1. Kramer’s (a.k.a Cramer’s) Rule • Component j of x = A-1b is • Form Bj by replacing column j of A with b.

2. Total Unimodularity • A square, integer matrix B is unimodular (UM) if its determinant is 1 or -1. • An integer matrix A is called totally unimodular (TUM) if every square, nonsingular submatrix of A is UM. • From Cramer’s rule, it follows that if A is TUM and b is an integer vector, then every BFS of the constraint system Ax = b is integer.

3. TUM Theorem • An integer matrix A is TUM 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 MCNFP constraint matrices are TUM.

4. General Form of the MCNF Problem

5. 2 1 3 Flow Balance Constraint Matrix Capacity Constraints Constraints in Standard Form

6. Shortest Path Problems • Defined on a Network • Nodes, Arcs and Arc Costs • Two Special Nodes • Origin Node s • Destination Node t • A path from s to t is an alternating sequence of nodes and arcs starting at s and ending at t: s,(s,v1),v1,(v1,v2),…,(vi,vj),vj,(vj,t),t

7. s=1, t=3 We Want a Minimum Length Path From s to t. 5 10 1 2 3 7 1 7 4 1,(1,2),2,(2,3),3 Length = 15 1,(1,2),2,(2,4),4,(4,3) Length = 13 1,(1,4),4,(4,3),3 Length = 14

8. Maximizing Rent Example • Optimally Select Non-Overlapping Bids for 10 periods

9. Shortest Path Formulation t d10 -7 -2 d9 -3 -7 -5 -2 0 0 -1 0 -4 d1 d2 d3 d4 d5 d6 d7 d8 -3 -6 s -1 -11

10. MCNF Formulation of Shortest Path Problems • Origin Node s has a supply of 1 • Destination Node t has a demand of 1 • All other Nodes are Transshipment Nodes • Each Arc has Capacity 1 • Tracing A Unit of Flow from s to t gives a Path from s to t

11. Maximum Flow Problems • Defined on a Network • Source Node s • Sink Node t • All Other Nodes are Transshipment Nodes • Arcs have Capacities, but no Costs • Maximize the Flow from s to t

12. Example: Rerouting Airline Passengers Due to a mechanical problem, Fly-By-Night Airlines had to cancel flight 162 - its only non-stop flight from San Francisco to New York. The table below shows the number of seats available on Fly-By-Night's other flights.

13. Max Flow from SF to NY = 2+2+5=9 2 D C 5 4 SF 4 NY 6 7 5 H A Formulate a maximum flow problem that will tell Fly-By-Night how to reroute as many passengers from San Francisco to New York as possible. (flow, capacity) (2,2) D C (4,5) (2,4) SF (2,4) NY (5,6) H A (7,7) (5,5)

14. MCNF Formulation of Maximum Flow Problems • Let Arc Cost = 0 for all Arcs • Add an infinite capacity arc from t to s • Give this arc a cost of -1

15. 2 D C 5 4 SF 4 NY 6 7 5 H A D C 5 4 SF 4 NY 6 5 H A Maximum-Flow Minimum-Cut Theorem • Removing arcs (D,C) and (A,NY) cuts off SF from NY. • The set of arcs{(D,C), (A,NY)} is an s-t cut with capacity 2+7=9. • The value of a maximum s-t flow = the capacity of a minimum s-t cut.