np-complete


Approximation Algorithm between two NP compete problems


Suppose that a O(n2)-time alpha-approximate algorithm exists for one of the two problems in each of the following pairs:
Vertex Cover and Independent Set
Independent Set and Clique
Max-Flow and Min-Cut
Does this guarantee that a O(n2)-time alpha-approximate algorithm exists for the other problem in the pair? I know that Clique reduces to Independent Set which in turn reduces to Vertex Cover.
Not necessarily, for two reasons.
First of all, NP reductions are generally not linear in complexity. Some of them are, but usually a problem of complexity n will reduce to some other NP problem of size n^3 or something. Even if we found a linear-time 3SAT algorithm, we wouldn't have found linear-time algorithms for all NP-hard problems -- just polynomial algorithms. So if by "similar" you mean "also n^2", not in general.
Secondly, approximations don't generally transfer. Because of the non-linear growth in complexity (that's a simplification of why, but it'll do), approximation guarantees generally don't survive the reduction process. As a result, while all NP-complete problems are in a sense comrades in exact solution hardness, they are far from it in approximation hardness.
In certain specific cases, approximations do transfer (and one of your examples -- left as an exercise for the reader -- most definitely transfers). But it's in no way guaranteed.

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