Algorithm Design

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Chapter 11 Approximation Algorithms
have S’ # 0 after selecting k centers, as it would have s ~ S’, and so it would
go on and select more than k centers and eventually conclude that k centers
cannot have covering radius at most r. This contradicts our choide of r, and
the contradiction proves that r(C) _ O. The goal is to find a set cover e so that the total
weight
is minimized. Note that this problem is at least as hard as the decision version
of Set Cover we encountered earlier; if we set al! wi = 1, then the minimum
11.3 Set Cover: A General Greedy Heuristic
weight of a set cover is at most k if and only if there is a collection of at most
k sets that covers U.
~ Designing the Algorithm
We will develop and analyze a greedy algorithm for this problem. The algorithm
will have the property that it builds the cover one set at a time; to choose
its next set, it looks for one that seems to make the most progress toward the
goal. What is a natural way to define "progress" in this setting? Desirable
sets have two properties: They have small weight wi, and they cover lots of
elements. Neither of these properties alone, however, would be enough for
designing a good approximation algorithm. Instead, it is natural to combine
these two criteria into the single measure wi/ISil--that is, by selecting Si, we
cover tSit elements at a cost of w i, and so this ratio gives the ,COSt per element
covered," a very reasonable thing to use as a guide.
Of course, once some sets have already been selected, we are only concerned
with how we are doing on the elements still 1eft uncovered. So we will
maintain the set R of remaining uncovered elements and choose the set Si that
minimizes wi/[S i ~ RI.
Greedy-Set-Cover:
Start with R= U and no sets selected
While R # 0 -
Select set S i that minimizes u~i/[S ~ N
Delete set S~ from R
EndWhile
Return the selected sets
As an example of the behavior of this algorithm, consider what it would do
on the instance in Figure 11.6. It would first choose the set containing the four
nodes at the bottom (since this has the best weight-to-coverage ratio, !/4). It
then chooses the set containing the two nodes in the second row, and finally
it chooses the sets containing the two individual nodes at the top. It thereby
chooses a collection of sets of total weight 4. Because it myopically chooses
the best option each time, this algorithm misses the fact that there’s a way to
cover everything using a weight of just 2 + 2~, by selectit~g the two sets that
each cover a fi~ column.
~4~ Analyzing the Algorithm
The sets selected by the algorithm clearly form a set cover. The question we
want to address is: How much larger is the weight of this set cover than the
weight ~v* of an optimal set cover?
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