CS 573: Graduate Algorithms, Fall 2011
HW 5 (due in class on Tuesday, November 29th)
This homework contains five problems.
Read the instructions for submitting homework on
the course web page
. In particular,
make sure
that you write the solutions for the problems on
separate sheets of paper. Write your name and netid on each sheet.
Collaboration Policy:
For this home work students can work in groups of up to three students
each. Only one copy of the homework is to be submitted for each group. Make sure to list all the
names/netids clearly on each page.
Note on Proofs:
Details are important in proofs but so is conciseness. Striking a good balance
between them is a skill that is very useful to develop, especially at the graduate level.
1. (20 pts) Consider the following scheduling problem.
There are
n
unitsized jobs that need
to be scheduled on
m
identical machines. Each job
j
has a release time
r
j
and a deadline
d
j
both of which are integers. A schedule assigns each job to a time slot [
t, t
+ 1] for some
integer
t
and to some machine such that no two jobs are assigned to the same slot on the
same machine.
If job
j
is completed by its deadline there is no penalty; if it completes at
time
C
j
> d
j
then it incurs a penalty of (
C
j

d
j
).
A job cannot be schedule before it is
release time. The goal is find a schedule for the jobs on the given machines so as to minimize
the total penalty.
Describe a polynomial time algorithm for this problem.
Be sure to pay
attention to the fact that your algorithm runs in polynomial time by examining carefully the
size of the input.
2. (20 pts) Reduce the following two problems to the problem of computing a minimumcost
perfect matching in a bipartite graph.
•
Given an undirected graph
G
= (
V, E
) and edge costs
c
:
E
→
R
find the minimumcost
2matching in
G
. A 2matching in
G
is an assignment
x
:
E
→
Z
+
such that
x
(
δ
(
v
)) = 2
for each
v
.
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 Fall '08
 Chekuri,C
 Algorithms, Optimization, Bipartite graph, polynomial time, maximum weight

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