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.

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.

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.

General Form of the MCNF Problem

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

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

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

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

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

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

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

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.

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)

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

Maximum-Flow Minimum-Cut Theorem SF 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.