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Many problems in computer science come in two flavors:

  • Optimization problem: "Find an object with the largest size".
  • Decision problem: "Given $n$, find an object with a size of at least $n$, or reply that such an object does not exist".

Given a solver for the optimization problem, the decision problem can be solved simply by checking if the optimal solution has a size of at least $n$. But this doesn't help if the optimization problem is NP-hard.

MY QUESTION IS: If we have a constant-factor approximation algorithm for the optimization problem, how can we use it for the threshold problem?

A possible way is: if the approximate solution has a size of at least $n$, return it; if it has a size of less than $n/c$ (where c is the approximation constant), return that an object of size $n$ does not exist; otherwise, return "I don't know". This may return a correct answer in many cases, but not in all cases.

Is there a better way?

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    $\begingroup$ Approximation algorithm for an optimization problem finds an answer at most $\alpha$ times worse than the optimal one. What does an approximation algorithm for a problem which only returns true or false do? Do you want that $\alpha$ to turn into probability? $\endgroup$
    – Karolis Juodelė
    Jan 1 '14 at 13:32
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    $\begingroup$ @KarolisJuodelė that's what I had in mind - create an algorithm that returns the correct answer with a provably high probability. Is this possible in general? $\endgroup$
    – Erel Segal-Halevi
    Jan 1 '14 at 14:21
  • $\begingroup$ @ErelSegalHalevi That seems impossible as long as you don't know anything about the distribution of the result within the error interval of the approximation algorithm. $\endgroup$
    – Raphael ♦
    Feb 19 '15 at 8:21
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First off, a technicality: optimization problems aren't NP-complete; only decision problems can be NP-complete (or in NP). Thus, it's only the decision problem (what you call the "threshold problem") that is in NP, or NP-complete.

On to the substance of your question. The answer is: You can't. In general, the approximation algorithm typically doesn't let you solve the decision problem. Think about it. We have many problems for where the decision problem is NP-complete, but where we have a polytime approximation algorithm. If the approximation algorithm let us quickly solve the decision problem (or the corresponding optimization problem), then we'd immediately have a polytime solution to a NP-complete (or NP-hard) problem, and thus a proof that P=NP. When you start with some assumption and it leads to a conclusion that is too good to be true, that's a sign that the assumption probably wasn't valid.

No, your "obvious answer" doesn't work. Decision problems aren't allowed to return "I don't know". No, there isn't a better way (assuming the problem is NP-hard, and assuming $P\ne NP$).

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