Algorithm Design

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Chapter 11 Approximation Algorithms
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wo sets can be used to ~
over everything, but the|
reedy algorithm doesn’t|
nd them. J
Figure 11.6 An instance of the Set Cover Problem where the weights of sets are either
1 or 1 + e for some small e > 0. The greedy algorithm chooses sets of tot.a1 weight 4,
rather than the optimal solution of weight 2 + 2e.
As in Sections 11.1 and 11.2, our analysis will require a good lower bound
on this optimum. In the case of the Load Balancing Problem, we used lower
bounds that emerged naturally from the statement of the problem: the average
load, and the maximum iob size. The Set Cover Problem wil! turn out to be
more subtle; "simple" lower bounds are not very useful, and instead we will
use a lower bound that the greedy algorithm implicitly constructs as a byproduct.
Recall the intuitive meaning of the ratio wi/ISi ~ RI used by the algorithm; it
is the "cost paid" for covering each new element. Let’s record this cost paid for
11.3 Set Cover: A General Greedy Heuristic
element s in the quantity c s. We add the following line to the code immediately
after selecting the set S t.
Define Cs=Wi/]Sig~ R[ for all s~SiN R
The values c s do not affect the behavior of the algorithm at all; we view
them as a bookkeeping device to help in our comparison with the optimum
w*. As each set S i is selected, its weight is distributed over the costs c s of the
elements that are ~ewly covered. Thus these costs completely account for the
total weight of the set cover, and so we have
(11.9) If ~ is the set cover obtained by Greedy-Set-Cover, then ~s~e wi =
~s~u Cs.
The key to the analysis is to ask how much total cost any single set S k
can account for--in other words, to give a bound on ~,s~sk Cs relative to the
weight w k of the set, even for sets not selected by the greedy algorithm. Giving
an upper bound on the ratio
~k
that holds for every set says, in effect, "To cover a lot of cost, you must use a lot
of weight." We know that the optimum solution must cover the full cost ~s~u Cs
via the sets it selects; so this type of bound will establish that it needs to use
at least a certain amount of weight. This is a lower bound on the optimum,
just as we need for the analysis.
Our analysis will use the harmonic function
n
H(n) = ~ 1
i---1 l
To understand its asymptotic size as a function of n, we can interpret it as a
sum approximating the area under the curve y = 1Ix. Figure 11.7 shows how
it is naturally bounded above by 1 + f{ nl~
dx = 1 + In n, and bounded below
. rn+l 1
by dl ~ ax = ln(n + !). Thus we see that H(n) = ® (lnn).
Here is the key to establishing a bound on the performance of the algo-
(11.10) For every set Sk, the sum ~s~sk Cs is at most H (ISk[ ) ¯
Proof. To simplify the notation, we will assume that the elements of S~ are
sa]. Furthermore,
let us assume that these elements are labeled in the order in which they are
assigned a cost cs~ by the greedy algorithm (with ties broken arbitrarily). There
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