# HW 2: due Thurs, February 3 ```COMP 260: Advanced Algorithms
Tufts University
Prof. Lenore Cowen
Spring 2011
HW 2: due Thurs, February 3
1. A certain commodity is produced at two factories x1 and x2 . The commodity is to be shipped to markets y1 , y2 and y3 through the network
shown below. What is the maximum ammount that can be shipped
from the factories to the markets?
y1
18
x1
4
7
a
5
3
e
24
8
7
16
y2
13
12
24
c
2
7
22
2
b
19
d
15
6
x2
4
y3
2. Let G be a directed graph with source s and sink t. Suppose the
capacities are specified not on the edges of G but 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 s to t in 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 KóningEgeváry theorem, i.e. if G is bipartite, then the size of the maximal
matching is equal to the size of the minimum vertex cover.
```