6
Network Flow Problems
Yet another area of rich applications of graphs (actually digraphs) deals with so called networks
and commodity flows in them.
This time, the main optimization task is to maximize a flow from the designated source to
the designated
10
The Minimum-Cost Flow Problem
The remaining lectures will be concerned with optimization problems on networks, in
particular with flow problems.
10.1
Networks
A directed graph, or network, G = (V, E) consists of a set V of vertices and a set
E V V of e
Minimum Cost Flow
Notations:
Directed graph G = (V, E)
Let u denote capacities
Let c denote edge costs.
A flow of f (v, w) units on edge (v, w) contributes cost c(v, w)f (v, w) to
the objective function.
Different (equivalent) formulations
Find the m
Network Flow III - Applications
4/5/05
1
Previous Lecture
u
v
12
S
16
s
12 /12
u
T
v
15/16
19/20
20
10
4
t
7
9
s
4/10
1/4
0 /9
8/13
4 /4
13
4
x
w
t
7 /7
w
11/14
x
14
flow network
valid flow
Theorem 1.1 Given a network flow with n vertices and m edges, one
Makeup 3:305 in 36-144 on 10/15
1
Min-Cost Flow
Many different max-flows in a graph. How compare?
cost c(e) to send a unit of flow on edge e
P
find max-flow minimizing
c(e)f (e)
costs may be positive or negative!
note: pushing flow on cost c edge crea
Minimum Cost Flow
Algorithms and Networks
This lecture
The minimum cost flow problem: statement
and applications
The cycle cancelling algorithm
A polynomial time variant of cycle
cancelling
The successive shortest paths algorithm
2
Algorithms and Netw
Massachusetts Institute of Technology
6.854J/18.415J: Advanced Algorithms
David Karger
Handout 20
Wednesday, November 9, 2005
Problem Set 9 Solutions
Problem 1.
(a) Let S be an independent set in G. If in the product graph we choose from each Gv ,
where v