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CS 7250, Approximation Algorithms
Homework 4
Thu, April 5, 2011
Due Tue, April 12, 2011
Problem 1
The performance guarantee
α
OPT,
α >
0
.
878, for approximating MAXCUT was assuming that
the vector program (26.2) on page 256 of Vazirani’s book can be solved optimally in polynomial
time. In reality, such vector programs cannot be solved optimally in polynomial time, but they
can be approximated to any desired degree of accuracy in polynomial time.
Therefore, if
~
a
i
, 1
≤
i
≤
n
, is an oprimal solution for (26.2), suppose that we have instead an
approximation
~
b
i
, 1
≤
i
≤
n
, with the guarantee that, for all
i
, (a)(1

±
)

~
a
i
 ≤ 
~
b
i
 ≤
(1 +
±
)

~
a
i

and (b)
~
b
i
lies inside a cone of angle
φ
±
around the direction of
~
a
i
.
How would that aﬀect the statement of Lemma 26.9 ? You may assume that 0
≤
± <<
1
/
100 and
0
≤
φ
±
<< π/
100.
Problem 2
(a) In Vazirani’s book, for MAXCUT, Theorem 26.11 shows how to obtain a ”high probability
statement” from Lemma 26.9. Obtain a similar statement for MAS2SAT, starting from Lemma
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 Algorithms

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