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. Flow Balance Constraint Matrix1 2 3
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. 1 5 2 10 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
d10 -7 -2 d9 -3 -7 -5 -2 -1 -4 d1 d2 d3 d4 d5 d6 d7 d8 -3 s -6 -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 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 7
(flow, capacity)
(2,2)
(4,5) D C
(2,4)
Max Flow from SF to NY
= 2+2+5=9 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. Maximum-Flow Minimum-Cut TheoremSF D H C A NY 5 6 2 4 7
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.