# Super-polynomial time approximation algorithms for optimization problems

This is motivated by my previous question, Super-polynomial time approximation algorithms for MAX-3SAT. For many optimization problems, for each one we have inapproximability lower bound $\alpha$ assuming some widely believed complexity theoretic conjecture. In other words, there is no polynomial time algorithm for such optimization problems with approximation ratio better than some $\alpha$ (different ratio $\alpha$ for each problem).

Are there optimization problems for which we can achieve approximation ratio better than $\alpha$ if we allow super-polynomial time algorithms? Can we achieve better approximation ratios using quasi-polynomial time algorithms ( $n^{O(\log n)}$) or even using sub-exponential time algorithms($2^{o(n)}$)?

I would appreciate a survey of such results.

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One example is Maximum Independent Set. It is NP-hard to approximate the problem with ratio $n^{1-\epsilon}$ (Zuckerman, 2007). However, Bourgeois et al. (2011) give a simple $n^{1/2}$-approximation algorithm with running time $O^*(2^{\sqrt{n} \log n})$. Here, $n$ denotes the number of vertices of the input graph and the $O^*$-notation hides polynomial factors.
Another example is the Bandwidth problem. Dunagan & Vempala (2001) design an algorithm with approximation ratio $O(\log^3 n)$ and running time $O(n^{\log n})$. To my knowledge, the best known polynomial time approximation has approximation ratio $O(\log^3 n (\log \log n)^{1/4})$ (Lee, 2009); however, no $\omega(\sqrt{\log n / \log \log n})$ lower bound is known for the best approximation ratio achievable in polynomial time.
Actually, for bandwidth it seems that there is a super-constant inapproximability bound, especially if one is willing to assume that NP cannot be solved in quasi-polynomial time. The bound, given in "Hardness results for approximating the bandwidth" by Dubey, Feige, Unger (JCSS 2011), is $c\sqrt{\log n/\log\log n}$. – Michael Lampis Feb 20 '13 at 12:34