Vertex Cover:
Data Structure:
n: number of vertices
m[1..n][1..n]: Adjacency matrix,
B: number giving target length of dominating set
Length
O(n^2) elements
We don’t need the exact number. We just need a polynomiallyrelated estimate.
Witness
w: Finite but unbounded list of labels for vertices.
Note that this structure will be stored on the left
of our Turing machine’s tape by the oracle. In
our model, our verification algorithm must treat this as an essentially
random
string of symbols
from the tape alphabet we will assume that all such strings will map to some instance of our
witness data structure in an efficient (read: polynomial time) manner, and that there is an oracle
string for every instance of our witness data structure.
In short, just give a simple data structure and let the oracle fill in the fields. Do not put any
preconditions on the witness, such as “the length of the set is less than k.” If the oracle does too
much work, we could have a polynomial verification algorithm for a problem that is not NP!
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 Fall '10
 Dutton
 verification algorithm

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