Algorithm Design

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Chapter 11 Approximation Algorithms
11.4 The Pricing Method: Vertex Cover
We now turn to our second general technique for designing approximation
algorithms, the pricing method. We wil! introduce this technique by considering
a version of the Vertex Cover Problem. As we saw in Chapter 8, Vertex
Cover is in fact a special case of Set Cover, and so we will begin this section
by considering the extent to which one can use reductions in the design of
approximation algorithms. Following this, we will develop an algorithm with
a better approximation guarantee than the general bound that we obtained for
Set Cover in the previous section.
~ The Problem
Recall that a vertex cover in a graph G = (V, E) is a set S __ V so that each
edge has at least one end in S. In the version of the problem we consider here .....
each vertex i ~ V has a weight FO i >_ O, with the weight of a set S of vertices
denoted Lv(S) = ~i~S LVi. We would like to find a vertex cover S for which tv(S)
is minimum. When all weights are equal to 1, deciding if there is a vertex cover
of weight at most k is the standard decision version of Vertex Cover.
Approximations via Reductions? Before we work on developing an algorithm,
we pause to discuss an interesting issue that arises: Vertex Cover is
easily reducible to Set Cover, and we have iust seen an approximation algorithm
for Set Cover. What does this imply about the approximability of Vertex
Cover? A discussion of this question brings out some of the subtle ways in
which approximation results interact with polynomial-time reductions.
First consider the special case in which all weights are equal to !--that
is, we are looking for a vertex cover of minimum size. We wil! call this the
urtweighted case. Recall that we showed Set Cover to be NP-complete using a
reduction from the decision version of unweighted Vertex Cover. That is,
Vertex Cover