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  • $\begingroup$ "An oracle for a problem P is a (hypothetical) subroutine that can solve instances of P in constant time." Must an oracle take only constant time? $\endgroup$ Commented Oct 11, 2012 at 4:17
  • $\begingroup$ @Tim Of course there are books, I listed a few in the comments of another answer $\endgroup$ Commented Oct 21, 2012 at 8:33
  • $\begingroup$ @Tim Regarding the oracle: If you have found/conceived a reduction $A\le_T B$ between two problems $A$ and $B$ you have reduced the problem of finding an efficient algorithm for $A$ to finding an efficient algorithm for $B$. Or in other words the reduction tells you that in order to solve $A$ you can use $B$. It is like using a subroutine for $B$ in an algorithm for $A$. However the problems $A$ and $B$ are often problems where we don’t know efficient solutions. And in case of Turing-reducibility we even use it in cases where the problems involved aren’t decidable at all. $\endgroup$ Commented Oct 21, 2012 at 9:58
  • $\begingroup$ @Tim Thus $B$ is an unknown subroutine. It has become a custom in complexity theory to call the hypothetical algorithm for $A$ derived from the reduction as an algorithm with oracle $B$. Calling the unknown subroutine for $B$ an oracle just expresses that we can’t hope to find an efficient algorithm for $B$ just as we can’t hope to obtain an oracle for $B$. This choice is somewhat unfortunate, as it connotes a magical ability. The cost for the oracle should be $|x|$ as a subroutine has at least to read the input $x$. $\endgroup$ Commented Oct 21, 2012 at 9:59
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    $\begingroup$ An excellent answer all around; the only thing I would add (coming at it now via another question) is that the 'optimization direction' is a needless bit of complexity and for concreteness we can always presume that the objective function $Z$ is to be maximized; if the intention is to minimize, then we can just define a new objective function $Z'=-Z$ and rewrite all the minimization of $Z$ as maximization of $Z'$. $\endgroup$ Commented Aug 21, 2013 at 15:28