I'm looking for examples of hard optimization problems, for which we have an optimal approximation (not that this is not the same as $PTAS$, as we require a completely tight approximation, and not $1+\epsilon$-multiplicative approximation), in a sense defined as follows:
Formally, let $L\subseteq\Sigma^*\times \mathbb N$ be the decision version of some minimization (or maximization, flipping the definitions) problem, i.e. $$\forall n\leq m:(w,n)\in L\implies (w,m)\in L$$
Which examples of such known languages $L$ is $NP$-hard, but if we are allowed of a minimal relaxation, it is poly time solvable.
By minimal relaxation I mean that there exists a TM such that given an input $(w,n)$:
- The machine accepts if $(w,n+1)\in L$.
- The machine rejects if $(w,n-1)\not \in L$.
- The machine may act arbitrarily on other cases.
An example for such problem is the Degree-constrained spanning tree, where the problem is finding an MST with minimal maximum vertex degree.
Which other languages are NP-hard but subject to optimal approximation in poly time?