HW 2: due Thurs, February 3
1. A certain commodity is produced at two factories x1and x2. The com-
modity is to be shipped to markets y1,y2and y3through the network
shown below. What is the maximum ammount that can be shipped
from the factories to the markets?
x1
x2
ab
cd
y1
y2
y3
5
7
18 4
2
7
12
15
4
6
24
e
24
22
19
3713
16
8
2
2. Let Gbe a directed graph with source sand sink t. Suppose the
capacities are specified not on the edges of Gbut on the vertices (other
than s,t); for each vertex there is a fixed limit on the total flow through
it. There is no restriction on flow through the edges. Show how to use
the ordinary network flow theory to determine the maximum capacity
of a feasible flow from sto tin the vertex-capacitated graph G.
3. A vertex cover of a graph is a set of vertices C, such that every edge
has at least one endpoint in C. Use network flows to prove the K´oning-
Egev´ary theorem, i.e. if Gis bipartite, then the size of the maximal
matching is equal to the size of the minimum vertex cover.