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AlgorithmsComplexity Homework8
Prateek Sharma  09305910
November 10, 2009
1
Problem 1  3Coloring
We use similar techniques like we did for the MAX3SAT randomized algorithm.
Pick any vertex and assign a color randomly. Since there are 3 colors, the
probability is 2
/
3 that a random assignment of colors to these two vertices is
safe(legal). That is the 2 colors need to be diﬀerent. Let
X
be event that the
coloring of 2 vertices is legal (diﬀ colored). Let
Y
=
¯
X
be the event that they
are assigned same color.
Pr
(
Y
) = 1
/
3. Hence
Pr
(
X
) = 1

Pr
(
¯
X
).
Each edge has the probability of 2
/
3 that its associated vertices are safely
colored.
Consider OPT, the optimal 3coloring algorithm. Let
OPT
be the number
of correctly colored vertices. Then ,
OPT
≤
E
.
The randomized algorithm described above. Let the number of edges cor
rectly colored (by that we really mean that the endpoints are diﬀerently colored)
be
N
. Then expected number of edges returned by the randomized algorithm
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 Summer '09
 PROF.RANADE
 Algorithms

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