1. Kramer’s (a.k.a Cramer’s) Rule Component j of x = A -1 b is Form B j 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. 1 2 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,v 1 ),v 1,(v 1,v 2 ),…,(v i,v j ),v j,(v j,t),t

7. 123 4 510 7 7 1 s=1, t=3 1,(1,2),2,(2,3),3Length = 15 1,(1,2),2,(2,4),4,(4,3)Length = 13 1,(1,4),4,(4,3),3Length = 14 We Want a Minimum Length Path From s to t.

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

9. d1d2 -2 d3d4d5d6d7d8 d9 d10 -7 -2 -11 -6 00 -3 -7 0-4 -5 -3 s t Shortest Path Formulation

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. 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. SF D H C A NY 5 6 2 4 5 4 7 SF D H C A NY (4,5) (2,2) (2,4) (5,5) (2,4) (7,7) Max Flow from SF to NY = 2+2+5=9 (flow, capacity) (5,6)

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. 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. SF D H C A NY 5 6 2 4 5 4 7 SF D H C A NY 5 6 4 5 4