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I just started reading Vazirani's book "Approximation Algorithms". It is legally available online here.

On page 5 (23 in the pdf), it says that the following decision problems are in NP:

  • Is the size of the minimum vertex cover in $G$ at most $k$?
  • Is the size of the maximum matching in $G$ at least $l$?

I can't see why this is true. If, say, the first problem is in NP, then each valid instance (graph) has a Yes certificate which proves that the graph has a minimum vertex cover of size at most $k$.

But, how can one deduce from a certificate something regarding the size of the minimum vertex cover?

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  • $\begingroup$ This book is still on my todo list, but I have to comment you probably will want an (lower level) book on optimization as reference as it seems to assume you're comfortable with non-approximation algorithms already. You might want to take a look at this freely available pdf by Alexander Schrijver: homepages.cwi.nl/~lex/files/dict.pdf Chapter 6 covers some of the main concepts in the introduction of Vazirani a more accesable way I think. $\endgroup$ – Thomas Bosman Aug 17 '15 at 21:49
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Any vertex cover of size $k$ is a witness that the smallest one has at most size $k$. Hence, the questions

Is there a vertex cover of size at most $k$?

and

Does a smallest vertex have size at most $k$?

are equivalent.

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    $\begingroup$ Additional exercise, related to the question: Given an algorithm which decides if a vertex cover of a given size exists, how many times would you need to run this algorithm to find the size of the minimal cover? $\endgroup$ – Pseudonym Aug 16 '15 at 21:34

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