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In theorem 4 of Approximability of Minimum-weight Cycle Covers Bodo Manthey proves that:

Then no approximation algorithm for $\operatorname{Min-L-DCC}$ achieves an approximation ratio of $o(n)$, where n is the number of vertices of the input graph.
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Thus, no recursive algorithm can achieve an approximation ratio of $o(n)$

Questions:

  • is my conclusion right that there is general definition of approximation algorithms in the context of $NP$ completeness and that these algorithms are certain recursive algorithms that construct the solution for a problem with $n$ vertices from solutions for subsets of smaller size?
  • Where can I find an exact definition of that class of approximation algorithms?
Improve this question
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    $\begingroup$ Min-LDCC is an optimization problem, a class of problems for which there is indeed a rigorous and general definition of approximation algorithm. NP is a class of decision problems, but there is a defn of "NP optimization problem" (where an associated decision problem is in NP). Vazirani's textbook Approximation Algorithms should have it. $\endgroup$ Commented Jul 3, 2022 at 16:52

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