# Slightly Superexponential Parameterized Problems

## TL;DR

This paper investigates parameterized problems with superexponential running times and proves that for some, the dependence on the parameter cannot be improved to single exponential, highlighting inherent computational complexity limits.

## Contribution

The paper demonstrates that for certain parameterized problems with $f(k)=k^{O(k)}$, the superexponential dependence on $k$ is optimal and cannot be reduced to single exponential.

## Key findings

- Proves superexponential dependence is optimal for some problems.
- Shows that $f(k)=k^{O(k)}$ cannot be improved to $c^k$ for these problems.
- Highlights inherent complexity barriers in parameterized algorithms.

## Abstract

A central problem in parameterized algorithms is to obtain algorithms with running time $f(k)\cdot n^{O(1)}$ such that $f$ is as slow growing function of the parameter $k$ as possible. In particular, a large number of basic parameterized problems admit parameterized algorithms where $f(k)$ is single-exponential, that is, $c^k$ for some constant $c$, which makes aiming for such a running time a natural goal for other problems as well. However there are still plenty of problems where the $f(k)$ appearing in the best known running time is worse than single-exponential and it remained ``slightly superexponential'' even after serious attempts to bring it down. A natural question to ask is whether the $f(k)$ appearing in the running time of the best-known algorithms is optimal for any of these problems. In this paper, we examine parameterized problems where $f(k)$ is $k^{O(k)}=2^{O(k\log k)}$ in the best known running time and for a number of such problems, we show that the dependence on $k$ in the running time cannot be improved to single exponential. (See paper for the longer abstract.)