- cc.complexity-theory optimization examples
- Updated Sat, 11 Jun 2022 10:22:26 GMT

Are there any known natural examples of optimization problems for which it is much easier to produce an optimal solution than to evaluate the quality of a given candidate solution?

For the sake of concreteness, we may consider polynomial-time solvable optimization problems of the form: "given x, minimize $f(x, y)$", where $f:\{0,1\}^*\times\{0,1\}^* \to \mathbb{N}$ is, say, #P-hard. Such problems clearly exist (for instance, we could have $f(x, 0) = 0$ for all $x$ even if $f$ is uncomputable), but I am looking for ``natural'' problems exhibiting this phenomenon.

In paper [1], there is a problem with the property that finding an optimal element takes polynomial time despite that computing the objective function values is NP-hard (it means that evaluating the quality of a given candidate solution is NP-hard as well).

[1] T.C.E.Cheng, Y.Shafransky, C.T.Ng. An alternative approach for proving the NP-hardness of optimization problems. European Journal of Operational Research 248 (2016) 5258.

Yakov Shafransky