Algorithm Design

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Chapter 8 NP and Computational Intractability
Our goal is to cover the edges in E, so we formulate an instance of Set
Cover in which the ground set U is equal to E. Each time we pick a vertex in
the Vertex Cover Problem, we cover all the edges incident to it; thus, for each
vertex i ~ V, we add a set Sic__ U to our Set Cover instance, consisting of all
the edges in G incident to i.
Sn
We now claim that U can be covered with at most k of the sets $1 .....
if and only if G has a vertex cover of size at most k. This can be proved very
easily. For if Sh ..... Si~ are ~ ! k sets that cover U, then every edge in G is
incident to one of the vertices il ..... i~, and so the set {i~ ..... ie} is a vertex
cover in G of size ~ _< k. Conversely, if {i~ ..... i~} is a vertex cover in G of size
~ _< k, then the sets S h ..... S h cover U.
Thus, given our instance of Vertex Cover, we formulate the instance of
Set Cover described above, and pass it to our black box. We answer yes if and
only if the black box answers yes.
(You can check that the instance of Set Cover pictured in Figure 8.2 is
actually the one you’d get by following the reduction in this proof, starting
from the graph in Figure 8.1.) m
Here is something worth noticing, both about this proof and about the
previous reductions in (8.4) and (8.5). Although the definition of