Algorithm Design

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Chapter 11 Approximation Algorithms
(11.34) For each item i we have v i W are not in
any solution, and hence can be deleted). We also assume for simplicity that
~-1 is an integer.
Knapsack-Approx (E) :
Set b = @/(2rt)) maxi vi
Solve the Knapsack Problem with values Oi (equivalently ~i)
Keturn the set S of items found
/.~ Analyzing the Algorithm
First note that the solution found is at least feasible; that is, ~7~ias wi 0 as claimed; but the dependence on the
desired precision ~ is not polynomial, as the running time includes ~-1 rather
than log ~-1. []
Finally, we need to consider the key issue: How good is the solution
obtained by this algorithm? Statement (!1.34) shows that the values 9i we used
are close to the real values vi, and this suggests that the solution obtained may
not be far from optimal. .,
then w~ have
Proof. Let S* be any set satisfying Y~,i~s* wi fi] = 2¢-lnb. Finally, the chain of inequalities
above says Y~i~s vi >- Y~.i~s fii - nb, and thus ~i~s vi > ( 2~-~ - 1)nb. Hence
nb < ~ Y~.iss vi for ~ _< !, and so
Evi