What are some problems in $\mathrm{P}$ with time complexity of high-degree polynomial?

Question

What are some problems that are in $\mathrm{P}$ but the best known algorithm has a high-degree polynomial ($\ge 3$) time complexity?

Answer 1

Concrete example: Thorup's $O(n^{120})$ time algorithm to recognize half-squares of planar bipartite graphs.

https://arxiv.org/abs/1804.05793

Parameterized problem: Some parameterized $NP$-complete problems have arbitrarily large polynomial time algorithms. Like, finding a $k$-clique in a graph can be solved by $\Theta(n^k)$ algorithm. And we do not know how to do much better. There might be some improvement but in general when the parameter increases the degree of the polynomial run-time bound increases also.

Computational geometry problems parameterized with the number of dimensions $d$ can be seen as a special case in this group.

Some other $NP$-complete problems (like Vertex Cover in the comment of kne below) have parameterized version with linear-time algorithm for each parameter value. So, it takes some careful investigations before naming a hard parameterized problems.

Answer 2

What are some problems that are in $\mathrm{P}$ but the best known algorithm has a high-degree polynomial ($\ge 3$) time complexity?

Polynomial time always makes me think of nested loops, such that $\ge \operatorname{O}\left(n^3\right)$ looks like "nest three-or-more loops".

So, optimizing $f\left(x_0 , \, x_1 , \, \dots , \,x_m \right) ,$ where:

• $f$ is a black-box function;

• each $x_i$ has $n$ distinct values; and

• $m \ge 2 .$

Answer 3

For an integer $k$, let $P_k$ be the following problem: The input is a program $p$ in your favourite programming language. The question is whether $p$ halts after at most $|p|^k$ steps, where $|p|$ is the input size (the length of the program $p$). Unless the chosen programming language is very unusual, every $P_k$ is decidable in polynomial time; yet for every exponent $e$, there is some $k$ such that $P_k$ requires $\Omega(n^e)$ time. This is an application of the time hierarchy theorem.

The above is an academic example. It is stronger than what you asked for: Not only the best known algorithm has a high exponent; all possible algorithms do. An example for a natural problem with a high-ish best-known exponent is primality. There, as far as I know, the exponent is $6$. (To be precise: This is the best known exponent for algorithm. The actual exponent might be lower. In fact, the first estimate for the same algorithm had an exponent of $12$.)