Massachusetts Institute of Technology
Handout 9
6.854J/18.415J: Advanced Algorithms
Wednesday, October 9, 2009
David Karger
Problem Set 5
Due: Wednesday, October 16, 2009.
Collaboration policy:
collaboration is
strongly encouraged
. However, remember that
1. You must write up your own solutions, independently.
2. You must record the name of every collaborator.
3. You must actually participate in solving all the problems. This is difficult in very large
groups, so you should keep your collaboration groups limited to 3 or 4 people in a given
week.
4.
No bibles. This includes solutions posted to problems in previous years.
Problem 1.
Linear Programming By Hand. Consider the following linear programming
problem:
minimize
cx
s.t.
x
1
+
x
2
≥
1
x
1
+ 2
x
2
≤
3
x
1
≥
0
x
2
≥
0
x
3
≥
0
For each of the following objectives
c
, give the optimum value and the set of optimum
solutions:
(a)
c
= (

1
,
0
,
0)
(b)
c
= (0
,
1
,
0)
(c)
c
= (0
,
0
,

1)
Problem 2.
You work for the ShortTerm Capital Management company and start the
day with
D
dollars. Your goal is to convert them to Yen through a series of currency trades
involving assorted currencies, so as to maximize the amount of Yen you end up with. You
are given a list of pending orders: client
i
is willing to convert up to
u
i
units of currency
a
i
into currency
b
i
at a rate of
r
i
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 Fall '09
 DavidKarger
 Algorithms, Optimization, Massachusetts Institute of Technology, Polytope, Stochastic matrix, David Karger

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