Section 2.9
Maximum Flow Problems
Minimum Cost Network Flows
Shortest Path Problems
Maximum Flow Problems
Reference: Network Flows by Ahuja,
Magnanti, and Orlin
Background
• In a capacitated network, we wish to send as much flow as possible
between two special nodes, a source node s and a sink node t,
without exceeding the capacity of any arc.
• The problem reportedly rose in relation to the rail networks of the
Soviet Union, during the 1950's. The US wanted to know how
quickly the Soviet Union could get supplies through its rail network
to its satellite states in Eastern Europe.
• In addition, the US wanted to know which rails it could destroy
most easily to cut off the satellite states from the rest of the Soviet
Union. It turned out that these two problems were closely related,
and that solving the max flow problem also solves the min cut
problem of figuring out the cheapest way to cut off the Soviet
Union from its satellites.
Additional reference: The Importance of Algorithms by Ibackstrom from www.topcoder.com
Assumptions
• The network is directed.
• All capacities are nonnegative integers.
• The network does not contain a directed path
from node s to node t composed only of
infinite capacity arcs.
• Whenever an arc (i,j) belongs to A, arc (j,i)
also belongs to A.
• The network does not contain parallel arcs.
Max Flow General Formulation
The standard LP formulation of the Maximum s-t Flow problem adds a dummy
Return arc, (t,s), to the network in question, and then maximizes the flow on this
arc. The capacity of the return arc is unbounded, and the rest of the formulation
consists of balance of flow and capacity constraints.
max X t , s
s.t.

j:( i , j ) A
X ij 

j:( j ,i ) A
X ji  0 i  V
0  X ij  uij for each (i, j )  A
Network Flow Model
Let V denote the set of nodes and A denote the set of arcs
in our network G. At each node i  V , let si denote the
supply at node i, and d j the demand at node j , and define
bi  si  di ; we shall assume that
 b  0,
iV
i
that is, the total amount of supply in the network equals
the total demand. The Minimum Cost Network Flow
Problem is given by
min

cij X ij
X ij 

( i , j ) A
s.t.

j:( i , j ) A
k :( k ,i ) A
X ij  0,
X ki  bi , i  V
(i, j )  A
Importance of Network Models
• Claim 2.1 – Nearly all network models can be
formulated as minimum cost network flow
problems
• Claim 2.2 – Any problem modeled as a
minimum cost network flow problem has an
optimal solution that has only integer values,
provided each supply and each demand has
integral value
College wants to know when to purchase a new copier over the next 6 years.
During those years that a copier is not purchased, maintenance must be performed
to keep machine running. Maintenance costs are shown below.
Age at Start of Year
Maint. Cost for Next Year
0
$2000
1
$3500
2
$6000
3
$9500
4
$14,500
At the start of this school year, a new copier is purchased at a cost of $10,000.
The cost (in today’s dollars) of purchasing a machine at the start of each of the
next 5 years is given below.
Year
Purchase Cost
1
$10,000
2
$13,000
3
$16,500
4
$20,000
5
$25,000
No trade-in value when a machine is replaced. Determine the years in which a new copier
should be purchased to minimize total cost (purchase + maintenance) for 5 years.
Costs
cij  (purchase price in year i )
 (maintenance costs during years i, i  1,..., j  1)
 (resale price in year j after j  i years of service).
c25  13  2  3.5  6  24.5
  12 15.5 21.5 30.5 43.5
   15 18.5 24.5 34 


c   
 18.5 22
28 






24
27.5


  



27 
Formulation
min

( i , j ) A
cij X ij
s.t.
X 12  X 13  X 14  X 15  X 16  1
X 23  X 24  X 25  X 26  X 12  0
X 34  X 35  X 36  X 13  X 23  0
X 45  X 46  X 14  X 24  X 34  0
X 56  X 15  X 25  X 35  X 45  0
 X 16  X 26  X 36  X 46  X 56  1
X ij  0,
(i, j )  A