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For example can an approximation algorithm call a subroutine which is solving a NP-Hard problem? (like say its trying to find the longest path in some graph as an intermediate step) Is that allowed?

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    $\begingroup$ What do you think? In general, any algorithm may use any other algorithm as subrouting, and even itself. $\endgroup$ – Raphael Oct 1 '14 at 20:08
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Depends. Typically, with "approximation algorithms" people understand algorithms that run in polynomial time and return a possibly non-optimal solution (with a formal guarantee on the solution quality). In this definition, no, you can't spend exponential time solving an NP-hard problem.

On the other hand, nothing is forcing you to consider polynomial time algorithms that return non-optimal solutions. In fact, it is interesting to consider algorithms that take exponential time, and still return non-optimal solutions. See for instance [1, 2].


[1] Cygan, Marek, Lukasz Kowalik, Marcin Pilipczuk, and Mateusz Wykurz. "Exponential-time approximation of hard problems." arXiv preprint arXiv:0810.4934 (2008).

[2] Cygan, Marek, Łukasz Kowalik, and Mateusz Wykurz. "Exponential-time approximation of weighted set cover." Information Processing Letters 109, no. 16 (2009): 957-961.

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It depends entirely on what complexity you want your algorithm to have. For example, if you're trying to approximate an EXP-hard problem then a polynomial algorithm that uses an NP oracle might be acceptable; if you're trying to approximate something that's only NP-hard, using an NP oracle more than once would be potentially more expensive than just solving the problem exactly.

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