Theoretical Computer Science
cc.complexity-theory np-hardness approximation-algorithms
Updated Wed, 01 Jun 2022 15:23:29 GMT

# NP-complete problems with optimal approximation in poly-time

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?

## Solution

One example is the chromatic index (edge-coloring number) of undirected graphs. By Vizing's theorem, the chromatic index either equals the max-degree $\Delta$ or equals $\Delta+1$.

• Vizing's proof gives a polynomial time (inductive) procedure for finding a coloring with $\Delta+1$ colors.
• Holyer has proved that deciding whether the chromatic index equals $\Delta$ is NP-complete.